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PREFACE. 


How  it  ever  came  to  pass  that  Arithmetic  should  be  taught 
to  the  extent  attained  in  the  grammar  schools  of  the  civilized 
world,  while  Geometry  is  almost  wholly  excluded  from  them, 
is  a  problem  for  which  the  author  of  this  little  book  has 
often  sought  a  solution,  but  with  only  this  result ;  viz.,  that 
Arithmetic,  being  considered  an  elementary  branch,  is  included 
in  all  systems  of  elementary  instruction  ;  but  Geometry,  being 
regarded  as  a  higher  branch,  is  reserved  for  systems  of  ad- 
vanced education,  and  is,  on  that  account,  reached  by  but  very 
few  of  the  many  who  need  it. 

The  error  here  is  fundamental.  Instead  of  teaching  the 
elements  of  all  branches,  we  teach  elementary  brandies  much  too 
exhaustively. 

The  elements  of  Geometry  are  much  easier  to  learn,  and 
are  of  more  value  when  learned,  than  advanced  Arithmetic ; 
and,  if  a  boy  is  to  leave  school  with  merely  a  grammar-school 
education,  he  would  be  better  prepared  for  the  active  duties  of 
life  with  a  little  Arithmetic  and  some  Geometry,  than  with 
more  Arithmetic  and  no  Geometry. 

Thousands  of  boys  are  allowed  to  leave  school  at  the  age 
of  fourteen  or  sixteen  years,  and  are  sent  into  the  carpenter- 
shop,  the  machine-shop,  the  mill-wright's,  or  the  surveyor's 
office,  stuffed  to  repletion  with  Interest  and  Discount,  but  so 


4  PEEFACE. 

utterly  ignorant  of  the  merest  elements  of  Geometry,  that  they 
could  not  find  the  centre  of  a  circle  already  described,  if  their 
lives  depended  upon  it. 

Unthinking  persons  frequently  assert  that  young  children 
are  incapable  of  reasoning,  and  that  the  truths  of  Geometry 
are  too  abstract  in  their  nature  to  be  apprehended  by  them. 

To  these  objections,  it  may  be  answered,  that  any  ordinary 
child,  five  years  of  age,  can  deduce  the  conclusion  of  a  syllo- 
gism if  it  understands  the  terms  contained  in  the  propositions  ; 
and  that  nothing  can  be  more  palpable  to  the  mind  of  a  child 
than  forms,  magnitudes,  and  directions. 

There  are  many  teachers  who  imagine  that  the  perceptive 
faculties  of  children  should  be  cultivated  exclusively  in  early 
youth,  and  that  the  reason  should  be  addressed  only  at  a 
later  period. 

It  is  certainly  true  that  perception  should  receive  a 
larger  share  of  attention  than  the  other  faculties  during  the 
'  first  school  years  ;  but  it  is  equally  certain  that  no  faculty  can 
be  safely  disregarded,  even  for  a  time.  The  root  does  not 
attain  maturity  before  the  stem  appears  ;  neither  does  the  stem 
attain  its  growth  before  its  branches  come  forth  to  give  birth 
in  turn  to  leaves  ;  but  root,  stem,  and  leaves  are  found  simul- 
taneously in  the  youngest  plant. 

That  the  reason  may  be  profitably  addressed  through  the 
medium  of  Geometry  at  as  early  an  age  as  seven  years  is  as- 
serted by  no  less  an  authority  than  President  Hill  of  Harvard 
College,  who  says,  in  the  preface  to  his  admirable  little  Georne- 
'  try,  that  a  child  seven  years  old  may  be  taught  Geometry  more 
easily  than  one  of  fifteen. 

The  author  holds  that  this  science  should  be  taught  in  all 


PREFACE.  5 

primary  and  grammar  schools,  for  the  same  reasons  that  apply 
to  all  other  branches.  One  of  these  reasons  will  be  stated  here, 
because  it  is  not  sufficiently  recognized  even  by  teachers.  It 
is  this :  — 

The  prime  object  of  school  instruction  is  to  place  in  the 
hands  of  the  pupil  the  means  of  continuing  his  studies  without 
aid  after  he  leaves  school.  The  man  who  is  not  a  student  of 
some  part  of  God's  works  cannot  be  said  to  live  a  rational  life. 
It  is  the  proper  business  of  the  school  to  do  for  each  branch  of 
science  exactly  what  is  done  for  reading. 

Children  are  taught  to  read,  not  for  the  sake  of  what  is  con- 
tained in  their  readers,  but  that  they  may  be  able  to  read  all 
through  life,  and  thereby  fulfil  one  of  the  requirements  of 
civilized  society.  'So,  enough  of  each  branch  of  science  should 
be  taught  to  enable  the  pupil  to  pursue  it  after  leaving  school. 

If  this  view  is  correct,  it  is  wrong  to  allow  a  pupil  to  reach 
the  age  of  fourteen  years  without  knowing  even  the  alphabet 
of  Geometry.  He  should  be  taught  at  least  how  to  read  it. 

It  certainly  does  seem  probable,  that  if  the  youth  who  now 
leave  school  with  so  much  Arithmetic,  and  no  Geometry,  were 
taught  the  first  rudiments  of  the  science,  thousands  of  them 
would  be  led  to  the  study  of  the  higher  mathematics  in  their 
mature  years,  by  reason  of  those  attractions  of  Geometry 
which  Arithmetic  does  not  possess. 


TO  THE  PROFESSIONAL  READER. 


THIS  little  book  is  constructed  for  the  purpose  of  instructing 
large  classes,  and  with  reference  to  being  used  also  by  teachers 
who  have  themselves  no  knowledge  of  Geometry. 

The  first  statement  will  account  for  the  many,  and  perhaps 
seemingly  needless,  repetitions  ;  and  the  second,  for  the  sugges- 
tive style  of  some  of  the  questions  in  the  lessons  which  develop 
the  matter  contained  in  the  review-lessons. 

Attention  is  respectfully  directed  to  the  following  points  :  — 

First  the  particular,  then  the  general.     See  page  25. 

Why  is  m  n  g  an  acute  angle  ? 

What  is  an  acute  angle  ? 

Here  the  attention  is  directed  first  to  this  particular  angle  ; 
then  this  is  taken  as  an  example  of  its  kind,  and  the  idea  gen- 
eralized by  describing  the  class.  See  also  page  29. 

Why  are  the  lines  ef  and  gh  said  to  be  parallel  ? 

AVhen  are  lines  said  to  be  parallel? 

Many  of  the  questions  are  intended  to  test  the  vividness  of 
the  pupil's  conception.  See  page  29. 

Also  page  78.  If  the  circumference  were  divided  into  360 
equal  parts,  would  each  arc  be  large  or  small? 

Many  of  the  questions  are  intended  to  test  the  attention  of 
the  pupil. 

The  thing  is  not  to  be  recognized  by  the  definition  ;  but 
the  definition  is  to  be  a  description  of  the  thing,  a  descrip- 
tion of  the  conception  brought  to  the  mind  of  the  pupil  by 
means  of  the  name. 


CONTENTS. 


PART  I. 

LINES 9 

POINTS 9 

CROOKED  LINES 10 

CURVED  LINES    .        . 11 

STRAIGHT  LINES 11 

OTHER  LINES       •         .         .         .         . 11 

POSITIONS  OF  LINES   .                 .         .  14 

ANGLES         .        .        .        .        .        .        .  .        .        .        .17 

RELATIONS  OF  ANGLES       . .  20 

ADJACENT  ANGLES 20 

VERTICAL  ANGLES       .         .         .         .         .  * 21 

KINDS  OF  ANGLES .23 

RIGHT  ANGLES 23 

ACUTE  ANGLES    . 24 

OBTUSE  ANGLES  .........;.  24 

RELATIONS  OF  LINES 27 

PERPENDICULAR  LINES *  .         .27 

PARALLEL  LINES 28 

OBLIQUE  LINES 28 

INTERIOR  ANGLES 30 

EXTERIOR  ANGLES 31 

OPPOSITE  ANGLES 32 

ALTERNATE  ANGLES 33 

PROBLEMS  RELATING  TO  ANGLES       .                 38 

POLYGONS 40 

TRIANGLES 44 

ISOSCELES  TRIANGLES  .        .        .48 


8  CONTENTS. 

PROBLEMS  RELATING  TO  TRIANGLES         .        .        .         .        .        .53 

QUADRILATERALS .55 

PARALLELOGRAMS        .        .        .         .        .        ...        .        .        .59 

COMPARISON  AND  CONTRAST  OP  FIGURES 62 

MEASUREMENT  OF  SURFACES 66 

PROBLEMS  RELATING  TO  SURFACES 71 

THE  CIRCLE  AND  ITS  LINES 73 

ARCS  AND  DEGREES 78 

PARTS  OF  THE  CIKCLE 82 


PART   II. 

AXIOMS    AND    THEOREMS. 

AXIOMS.    ILLUSTRATED 85 

THEOREMS.     ILLUSTRATED  ,     88 


OT  TOT 

MI7BRSIT7 


FIRST  LESSONS  IN  GEOMETRY. 


ZFIZR/ST. 
LESSON    FIRST. 

LINES. 

NOTE  TO  THE  TEACHER.  —  In  all  the  development-lessons,  the 
pupils  are  to  be  occupied  with  the  diagrams,  and  not  with  the  printed 
matter. 

See  Note  A,  Appendix. 

Refer  to  Diagram  1,  and  show  that 

What  are  here  drawn  are  intended  to  represent 

length  only. 

They  have  a  little  width,  that  they  may  be  seen. 
They  are  called  lines. 

A  line  is  that  which  has  length  only. 

POINTS 

Show  that 

Position  is  denoted  by  a  point. 

It  occupies  no  space. 

It  has  some  size,  that  it  may  be  seen. 

The  ends  of  a  line  are  points. 

A  line  may  be  regarded  as  a  succession  of  points. 

The  intersection  of  two  lines  is  a  point. 

A  point  is  named  by  placing  a  letter  near  it. 


10 


FIKST  LESSONS  IN  GEOMETRY. 


Diagram  1. 


A  point  may  be  represented  by  a  dot.     The  point 
is  in  the  center  of  the  dot. 

A  point  is  that  which  denotes  position  only. 

A  line  is  named  by  naming  the  points  at  its  ends. 
Read  all  the  lines  in  Diagram  1. 


CKOOKED    LINES. 
See  Note  B,  Appendix. 

Does  the  line  m  n  change  direction  at  the  point  1  ? 
At  what  other  points  does  it  change  direction  ? 
It  is  called  a  crooked  line. 

A  crooJfed  line  is  one  that  changes  direction  at  some 
of  its  points* 


LINES.  11 

CURVED    LINES. 

The  line  o  p  changes  direction  at  every  point. 
It  is  called  a  curved  line. 

A  curved  line  is  one  that  changes  direction  at  every 
point. 

STRAIGHT    LINES. 

Does  the  line  ij  change  direction  at  any  point  ? 
It  is  called  a  straight  line. 

A  straight  line  is  one  that  does  not  change  direction 
at  any  point, 

OTHER    LINES; 

The  line  q  r  winds  about  a  line. 
It  is  called  a  spiral  line. 
The  line  w  x  winds  about  a  point. 
It  also  is  called  a  spiral  line. 

A  spiral  line  is  one  that  winds  about  a  line  or  point. 

The  line  7  8  *  looks  like  waves. 
It  is  called  a  wave  line. 


What  kind  of  a  line  is  a  b  ? 
Why  ?     What  is  a  straight  line  ? 
What  kind  of  a  line  is  11  16  ? 
Why  ?     What  is  a  crooked  line  V 
What  kind  of  a  line  is  o  p  ? 
Why  ?     What  is  a  curved  line  ? 

*  To  be  read  seven,  eight,  not  seventy-eight. 


12  FIRST  LESSONS   IN   GEOMETRY. 

What  kind  of  a  line  is  s  t  ? 

Why? 

What  kind  o^  a  line  is  9  10  ? 

Why  ?     What  is  a  spiral  line  ? 

What  kind  of  a  line  \$>wx? 

Why? 


LESSON    SECOND. 

REVIEW. 

Read  all  the  straight  lines.     (DIAGRAM  2.) 

Why  is  m  n  a  straight  line  ? 

Define  a  straight  line. 

Read  all  the  crooked  lines. 

Why  is  7  8  a  crooked  line  ? 

Define  a  crooked  line. 

Read  all  the  curved  lines. 

Why  is  5  6  a  curved  line  ? 

What  is  a  curved  line  ? 

Read  all  the  wave  lines. 

Read  all  the  spiral  lines. 

Why  is  3  4  a^  spiral  line  ? 

Why  is  u  v  a  spiral  line  cc 

What  is  a  spiral  line  ? 


LINES. 


13 


Diagram    2. 


OF 


14 


FIEST  LESSONS  IN  GEOMETKY. 


Diagram    3. 

LESSON     THIRD. 

POSITIONS    OF    LINES. 

Let  the  pupils  hold  their  books  so  that  they  will  be  straight  up 
and  down  like  the  wall. 

VERTICAL    LINES. 

The  straight  line  a  b  points  to  the  center  of  the 

earth.     (DIAGRAM  3.) 
It  is  called  a  vertical  line. 
Name  all  the  vertical  lines. 

A  vertical  line  is  a  straight   line   that   points    to   the 
center  of  the  earth. 

HORIZONTAL    LINES. 

The  straight  line  o  p  points  to  the  horizon. 


LINES.  15 

t 

It  is  called  a  horizontal  line. 
Eead  all  the  horizontal  lines. 

A  horizontal  line  is  a  straight  line  that  points  to  the 
horizon. 

OBLIQUE    LINES. 

The  line  s  t  points  neither  to  the  center  of  the 

earth  nor  to  the  horizon. 
It  is  called  an  oblique  line. 
Read  all  the  oblique  lines. 

An  oblique  line  is  a  straight  line  that  points  neither 
to  the  horizon  nor  to  the  center  of  the  earth. 


E.  —  After  going  through  with  the  lemons  on  angles,  the 
pupils  may  be  told  that  oblique  lines  are  so  called  because  they  form 
oblique  angles  with  the  horizon. 


LESSON    FOURTH. 

REVIEW. 

Read  all  the  vertical  lines.     (DIAGRAM  4.) 

Why  is  q  r  a  vertical  line  ? 

What  is  a  vertical  line  ? 

Read  all  the  horizontal  lines. 

Why  is  5  6  a  horizontal  line  ? 

Define  a  horizontal  line. 

Read  all  the  oblique  lines. 

Why  is  s  t  an  oblique  line. 

What  is  an  oblique  line  ? 

NOTE.  —  Lines  that  point  in  the  same  direction  do  not  approach 
the  same  point. 


16 


FIBST  LESSONS  IN  GEOMETilY. 


Diagram  4. 


ANGLES. 


b     m 


Diagram 


LESSON    FIFTH. 

ANGLES. 

Do  the  lines  a  b  and  c  d  (DIAGRAM  5)  point  in  the 
same  direction?  (See  note,  page  15.) 

Then  they  form  an  angle  with  each  other. 

What  other  line  forms  an  angle  with  a  b  ? 

Which  of  the  two  lines  c  d,  ef,  has  the  greater  dif- 
ference of  direction  from  the  line  a  b  ? 

Then  which  one  forms  the  greater  angle  with  a  I  ? 

What  line  forms  a  still  greater  angle  with  the  line 
a  I? 


18  FIRST  LESSONS   IN   GEOMETRY. 

An  angle  is  the  difference  of  direction  of  two  straight 
lines. 

If  the  lines  a  I,  ef,  were  made  longer,  would  their 

direction  be  changed  ? 
Then  would  there  be  any  greater  or  less  difference 

of  direction  ? 
Then  would  the  angles  formed  by  them  be  any 

greater  or  less  ? 
Then  does  the  size  of  an  angle  depend  upon  the 

length  of  the  lines  that  form  it  ? 
If  the  lines  a  b,   e  f,  were   shortened,  would  the 

angle  formed  by  them  be  any  smaller  ? 
If  two  lines  form  an  angle  with  each  other,  and 

meet,  the  point  of  meeting  is  called  the  vertex. 
What  is  the  vertex  of  the  angle  formed  by  the 

lines  kj9  ij ?  —  ij,  i  I? 
An  angle  is  named  by  three  letters,  that  which 

denotes  the  vertex  being  in  the  middle.     Thus, 

the  angle  formed  by  kj,  ij,  is  read  Jcj  i,  or  ij  k. 
Bead  the  four  angles  formed  by  the  lines  m  n  and 

op. 
The  eight  formed  by  r  s,  t  u,  and  v  w. 


LESSON    SIXTH. 

REVIEW. 


Read  all  the  lines  that  form  angles  with  the  line 

a  b.     (DIAGRAM  6.)        ^ 
Which  of  them  forms  the  greatest  angle  with  it  ? 


ANGLES. 


Diagram  0. 


Which  the  least  ? 

Of  the  two  lines  c  d,  g  h,  which  form^  the  greater 
angle  with  ef? 

Eead  all  the  angles  whose  vertices  are  at  o  on  ij. 

Which  angle  is  the  greater,  I  o  m,  or  m  o  j  ?  —  iok, 
or  io  I? — loj,  or  m  o  j  ? 

Read  all  the  angles  formed  by  the  lines  v  w  and 
xy. 

Read  all  the  angles  above  the  line  n  p. 

Below  the  line  up.     Above  the  line  q  r. 

At  the  right  of  the  line  5  u. 

At  the  left.  *  At  the  right  of  the  line  &  t. 

At  the  left  of  the  line  s  t. 

Which  angle  is  the  greater,  n  1  3,  or  n  2  4  ? 

If  the  lines  x  y  and  v  w  were  lengthened  or  pro- 
duced, would  the  angles  v  z  x,  y  z  w  be  any 
greater  ? 

If  they  were  shortened,  would  the  angles  be  any 
less  ?  • 


20  FI11ST   LESSONS   IN   GEOMETRY. 

What  is  an  angle  ? 

Does  the  size  of  an  angle  depend  upon  the  length 
of  the  lines  which  form  it  ? 


Diagram   7. 

LESSON    SEVENTH. 

RELATIONS    OF    ANGLES. 

ADJACENT   ANGLES. 

Are  the  angles  a  e  c,  c  e  I  (DIAGRAM  7),  on  the  same 
side  of  any  line  ?  What  line  ?  * 

By  what  other  straight  line  are  they  both  formed  ? 

Then,  because  they  are  both  on  the  same  side  of 
the  same  straight  line  a  5,  and  are  both  formed 
by  the  second  straight  line  c  d,  they  are  called 
"adjacent  angles" 

The  angles  c  e  b,  I  e  d  are  both  on  the  same  side 
of  what  straight  line  ? 


RELATIONS   OF   ANGLES.  21 

They  are  both  formed  by  what  second  straight 
line? 

Then  what  kind  of  angles  are  they  ? 

Why  are  they  called  adjacent  angles  ? 

Read  the  adjacent  angles  below  the  line  a  b.  Be- 
low the  line  c  d. 

How  many  pairs  of  adjacent  angles  can  be  formed 
by  two  straight  lines  ? 

Read  all  the  adjacent  angles  formed  by  the  lines 
/  m  and  n  p. 

VEKTICAIi    ANGLES, 

Are  the  angles  a  e  c,  bed  formed   by  the   same 

straight  lines  ? 
Are-  they  adjacent  angles  ? 
They  are  called  "  vertical  angles." 
V&rtical  angles   are   angles  formed  by  the  same 

straight  lines,  but  not  adjacent  to  each  other. 
Read  the  other  pair  of  vertical  angles  formed  by 

the  lines  a  b,  c  d. 
Read  all  the  vertical  angles  formed  by  the  lines 

f  0,  i  h.     By  I  m,  up. 

Why  are  the  angles  /  o  n,  n  o  m  adjacent  angles  ? 
Why  are  the  angles  / o  n,p  o  m  vertical  angles? 


22 


FIKST  LESSONS  IN  GEOMETRY. 


Diagram  8. 


LESSON    EIGHTH. 

REVIEW. 

Eead  the  pairs  of  adjacent  angles  above  the  line 

a  b.     (DIAGRAM  8.) 
Why  are  they  adjacent  ? 
What  are  adjacent  angles  ? 
Read  the  adjacent  angles  below  the  line  a  b. 
On  the  right  of  the  line  c  d.     On  the  left. 
How  many  pairs  of  adjacent  angles  are  formed  by 

the  intersection  of  two  lines. 
Head  the  pairs  of  adjacent  angles  formed  by  the 

lines  f  g  and  i  h. 


KINDS   OF  ANGLES.  23 

X-- 

Read  all  the  adjacent  angles  formed  by  the  lines 

/  m,  n  p. 
Read  all  the  pairs  of  vertical  angles  formed  by  tlie 

lines  a  b,  c  d. 

Why  are  c  e  b  and  a  e  d  called  vertical  angles  ? 
What  are  vertical  angles  ? 
Read  all  the  pairs  of  vertical  angles  formed  by  the 

lines  h  i,fg. 
How  many  pairs  of  vertical  angles  are  formed  by 

the  intersection  of  two  lines  ? 
Read  all  the  pairs  of  vertical  angles  formed  by  the 

lines  /  m,  n  p. 


LESSON    NINTH. 

KINDS     OF     ANGLES. 
BIGHT    ANGLES. 

What  do  we  call  the  angles  aoc,cob?  (DIAGRAM  9.) 
Are  they  equal  to  each  other? 
Then  they  are  called  right  angles. 

A  right  angle  is  one  of  two  adjacent  angles  that  are 
equal  to  each  other. 

Are  the  adjacent  angles,  c  o  b,  b  o  d  equal  to  each 

other  ? 

Then  what  are  they  called  ? 
Read  the  right  angles  below  the  line  a  b.     On  the 

left  of  c  d. 
Read  three  right  angles  whose  vertices  are  at  p. 


FIRST  LESSONS  IN   GEOMETRY. 


Diagram   9. 


ACUTE    ANGLES. 

Is  the  angle  m  p  q  greater  or  less  than  the  right 

angle  m  p  r  ? 
Then  it  is  called  an  acute  angle. 

An  acute  angle  is  one  which  is  less  than  a  right  angle. 

Read  four  acute  angles  whose  vertices  are  at  p.  ^ 

Acute  means  sharp. 

Why  is  r  p  s  an  acute  angle  ? 

What  is  an  acute  angle  ? 

OBTUSE    ANGLES. 

Is  the  angle  *m  p  s  greater  or  less  than  the  right 
angle  m  p  r? 


KINDS   OF  ANGLES.  25 

Then  it  is  called  an  obtuse  angle. 

An  obtuse  angle  is  one  which  is  greater  than  a  right 
angle. 

What  other  obtuse  angle  has  its  vertex  at  p  ? 
Obtuse  means  blunt. 

Read  three  obtuse  angles  whose  vertices  are  at  x. 
Acute  and  obtuse  angles  are  also  called  oblique 
angles. 


LESSON    TENTH. 

REVIEW. 

Read  all  the  right  angles  formed  by  the  lines  a  b 

and  c  d.     (DIAGRAM  10.) 
Why  are    the    adjacent    angles  c  e  b,  bed,  right 

angles  ? 

What  is  a  right  angle  ? 

Read  four  right  angles  whose  vertices  are  at  n. 
Which  is  the  greater,  the  right  angle  p  q  r,  or  the 

right  angle  t  s  u  ? 

Can  one  right  angle  be  greater  than  another  ? 
Read  six  acute  angles  whose  vertices  are  at  n. 
Why  is  m  n  g  an  acute  angle  ? 
What  is  an  acute  angle  ? 
Which  is  greater,  the   acute  angle  m  n  ff,  or  the 

acute  angle  lum? 

May  one  acute  angle  be  greater  than  another  ? 
What  three  acute  angles  are  equal  to  one  right 

angle  ? 


26 


FIBST  LESSONS  IN  GEOMETBY. 


Which  of  the  two  acute  angles  vfiv,  y  x  z  is  the 

greater  ? 

Read  four  obtuse  angles  whose  vertices  are  at  n. 
Why  is/ n  m  an  obtuse  angle? 
What  is  an  obtuse  angle  ? 
What  does  obtuse  mean  ?     Acute  ? 
By  what  other  name  are  both  called  ? 
Which  is  greater,  the  large  acute  angle  1  4  2,  or 

the  small  obtuse  angle  143? 
How  much    greater  than  the  right  angle  is  the 

obtuse  angle/  n  I? 
How  much  less  than  a  right  angle  isfn  i? 


RELATIONS   OF  LINES. 


27 


Diagram    11. 


LESSON    ELEVENTH. 

RELATIONS     OF    LINES. 
PERPENDICULAR    LINES. 

What  kind  of  angles  do  the  lines  a  b  and  c  d  make 
with  each  other  ?  (DIAGRAM  11.) 

Then  they  are  perpendicular  to  each  other. 

What  line  is  perpendicular  to  x  y  ? 

Why  is  it  perpendicular  to  it  ? 

What  line  is  perpendicular  to  z  1  ? 

When  is  a  line  said  to  be  perpendicular  to  another  ? 

Can  a  line  standing  alone  be  properly  called  a  per- 
pendicular line  ? 


28  FIRST  LESSONS   IN   GEOMETRY. 

What  two  lines  are  perpendicular  to  the  lines  r  s  ? 
Is    the    line  g  h  perpendicular  to   the    line  i  j  ? 

Why? 

What  other  line  is  perpendicular  to  the  line  //  ? 
Read  three  lines  that  are  perpendicular  to  the  line 

a  b. 

PARALLEL    LINES. 

Do  the  lines  Jc  Z,  m  n,  differ  in  direction  ?     Then  do 

they  form  any  angle  with 'each  other? 
They  are  said  to  be  parallel  to  each  other. 
Read  four  other  lines  that  are  parallel  with  Jc  I 
What  line  is  parallel  with  210? 
Why? 

Lines  are  parallel  ivith  each  other  when  they  do  not 
differ  in  direction. 

OBLIQUE    LINES. 

What  kind  of  angles  do  the  lines  u  t  and  8  9  form 

with  each  other  ? 
Then  they  are  said  to  be  oblique  to  each  other. 

Lines  are  oblique  to  each  other  when  they  form  oblique 
angles. 

See  Note  C,  Appendix. 


RELATIONS   OF   LINES. 
eg  o 


29 


Diagram    12. 


LESSON    TWELFTH. 

REVIEW. 

Read  five  lines  that  are  perpendicular  to  the  line 

a  b.     (DIAGRAM  12.) 
Five  that  are  perpendicular  to  c  d. 
Two  that  are  perpendicular  to  u  v9  and  meet  it. 

Three  that  do  not  meet  it. 

Why  are  o  p  and  m  n  perpendicular  to  each  other  ? 
When  are  lines  said  to  be  perpendicular  to  each 

other? 

Read  four  lines  that  are  parallel  with  ef. 
Why  are  the  lines  ef  and  g  h  said  to  be  parallel  to 

each  other  ? 


30 


FIRST  LESSONS   IN   GEOMETRY. 


When  are  lines  said  to  be  parallel  to  each  other  ? 

Read  four  lines  that  are  parallel  to  5  6. 

Four  that  are  parallel  to  op. 

Is  any  line  parallel  to  u  v  ? 

Can  a  single  line  be  properly  called  perpendicular  ? 

Parallel? 
If  two  lines  are  perpendicular  to  each  other,  what 

angle  do  they  form  ? 
If  parallel,  what  angle  ?     If  oblique  ? 


Diagram    13. 

LESSON     THIRTEENTH. 

RELATIONS   OF   ANGLES. 
INTERIOR    ANGLES. 

Is  the  angle  a  m  n  between  the  parallels,  or  outside 

of  them?     (DIAGRAM  13.) 
It  is  called  an  interior  angle. 


RELATIONS   OF  ANGLES.  ,°>1 

Read  three  other  interior  angles  between  the  same 

parallels. 
Why  is  b  m  n  an  interior  angle  ? 

An  interior  angle   is   one  that   lies  between  parallel 
lines. 

Read  the  interior  angles  between  the  parallel  lines 

g  h  and  k  I. 

Why  is  o  p  I  an  interior  angle  ? 
What  is  an  interior  angle  ? 

EXTERIOR    ANGLES. 

Is  the  angle  a  m  e  between  the  parallels,  or  outside 

of  them  ? 

It  is  called  an  exterior  angle. 
Read  three  other  exterior  angles  formed  by  the 

lines  a  b,  c  d,  and  ef. 
Why  is  the  angle  c  nf  an  exterior  angle  ? 

An  exterior  angle  is  one  that  lies  outside  of  the  paral- 
lels. 


LESSON    FOURTEENTH. 

REVIEW. 

Read  all  the  interior  angles  formed  by  the  lines 

&  by  c  d,  and  ef. 

Why  is  m  n  d  an  interior  angle  ? 
What  is  an  interior  angle  ? 
Read  all  the  exterior  angles  formed  by  the  same 

lines. 


32 


FIRST  LESSONS  IN  GEOMETRY. 


Why  is  d  nf  an  exterior  angle  ? 

What  is  an  exterior  angle  ? 

Kead  all  interior  angles  formed  by  the  lines  g  h,  Jc  I, 

and  ij. 
All  the  remaining  interior  angles  in  the  diagram. 

All  the  exterior  angles. 


u          w 


Diagram    14. 


LESSON    FIFTEENTH. 

RELATIONS    OF    ANGLES. 
OPPOSITE    ANGLES. 

Are  the  angles  e  m  b,6  m  n,  on  the  same  side  of  the 
intersecting  line  ef? 

Are  they  adjacent  ? 

Are  e  m  b,  m  n  d,  on  the  same  side  of  the  intersect- 
ing line  ef? 

Are  they  adjacent  ? 


RELATIONS   OF   ANGLES.  Co 

Then  they  are  called  opposite  angles. 

Opposite  angles  lie  on  the  same  side  of  the  intersecting 
line,  but  are  not  adjacent. 

Are  the  angles  e  m  b,f  n  d,  on  the  same  side  of  the 

intersecting  line  ? 
Are  they  adjacent? 
Then  are  they  opposite  ? 
Are  they  interior  or  exterior  angles  ? 
Then  they  are  "  opposite  exterior  angles." 
Why  are  they  exterior  ? 
Why  are  they  opposite  ? 
Are  the  angles  b  m  n,  m  n  d,  opposite  angles  ? 
Are  they  interior  or  exterior  angles  ? 
Then  they  are  " opposite  interior  angles" 
Why  are  they  opposite  ?     Why  interior  ? 
Read  the  opposite  exterior  angles  on  the  left  of 

the  line  ef. 

Read  the  opposite  interior  angles  on  the  same  side. 
Are  the  opposite  angles  e  m  a,  m  n  c,  both  exterior 

or  interior  ? 

Then  they  are  opposite  exterior  and  interior  angles. 
Read  two  pairs  of  opposite  exterior  and  interior 

angles  on  the  right  of  ef.     On  the  left. 

ALTEKNATE    ANGLES. 

Do  the  angles  b  m  n,  m  n  c,  lie  on  the  same  side  of 

the  intersecting  line  ef? 
Are  they  adjacent  to  each  other  ? 
Are  they  vertical  angles  ? 
Then  they  are  alternate  angles. 


34  FIRST  LESSONS   IN   GEOMETRY. 

Alternate  angles  lie  on  different  sides  of  the  intersect- 
ing line,  and  are  neither  adjacent  nor  vertical. 

Are  the  alternate  angles  b  m  n,  m  n  c,  exterior  or 
interior  ? 

Then  they  are  called  " interior  alternate  angles" 

Read  another  pair  of  interior  alternate  angles 
between  a  b  and  c  d. 

Are  the  angles  e  m  b,  c  nf,  alternate  angles?  Why? 

Are  they  exterior  or  interior  ? 

Then  what  may  they  be  called  ? 

Head  another  pair  of  exterior  alternate  angles. 

Why  are  e  m  a,  d  nf,  alternate  angles?  Why  exte- 
rior alternate  ? 


DELATIONS  OF  ANGLES.  35 


LESSON    SIXTEENTH. 

REVIEW. 

Read  the  exterior  opposite  angles  on  the  right  of 

the  line  ef.     (DIAGRAM  14.) 
On  the  left.     On  the  right  of  r  s.     On  the  left. 
Why  are  e  m  a,  c  nf,  exterior  angles  ? 
Why  are  they  opposite  angles  ? 
What  are  opposite  angles  ? 
Read  the  interior  opposite  angles  on  the  right  of 

the  intersecting  line  ef. 

On  the  left  of  it,    On  the  right  of  r  s.    On  the  left. 
Bead  the  interior  alternate  angles  formed  by  the 

lines  a  b,  c  d,  and  ef. 
Which  pair  are  acute  angles  ? 
Which  pair  are  obtuse  angles  ? 
Why  are  b  m  n,  m  n  c,  interior  angles  ?     Why  alter- 
nate ?     What  are  alternate  angles  ? 
Read   the    exterior  alternate  angles  of   the  same 

lines. 
Read  the  acute  interior  alternate  angles   of  the 

parallels  t  u,  v  w.     The  obtuse. 
The  acute  exterior  alternate  angles.     Obtuse. 
Read  the  pair  of  opposite   exterior  angles  on  the 

right  of  the  line  ef.     On  the  left. 
On  the  right  of  r  s.     On  the  left. 


ftr  W  *»  i?rm«*V 


FIRST  LESSONS  IN   GEOMETRY. 


Diagram    15. 


'  LESSON    SEVENTEENTH. 

REVIEW. 

Read  thirteen  or  more  angles  whose  vertices  are 

at  0.     (DIAGRAM  15.) 
Read  four  obtuse  angles. 
Read  two  right  angles. 

What  three  acute  angles  equal  one  right  angle  ? 
Which  is  greater,  the  right  angle  4,  or  the  right 

angle  5  ? 

The  obtuse  angle  6,  or  the  acute  angle  7  ?  - 
Read  twelve  pairs  of  adjacent  angles  formed  by 

the  lines  w  x,  &c, 


RELATIONS    OF   ANGLES.  37 

Read  six  pairs  of  vertical  angles  formed  by  the 

same  lines. 
Read  all  the  interior  angles  formed  by  the  lines  ij, 

Jc  Z,  and  m  n. 
Read  all  the   exterior  angles  formed  by  the  same 

lines. 

Two  pairs  of  opposite  exterior  angles. 
Two  pairs  of  opposite  interior  angles. 
Four  pairs  of  opposite  exterior  and  interior  angles. 
Two  pairs  of  alternate  interior  angles. 
Two  pairs  of  alternate  exterior  angles. 
Why  are  the  angles  i  o  m,  m  oj,  called  adjacent? 
What  are  adjacent  angles  ? 
What  kind  of  an  angle  is  i  o  m  ?     Why  ? 
What  is  an  acute  angle  ? 
What  kind  of  an  angle  is  m  o  j  ?     Why  ? 
What  is  an  obtuse  angle  ? 
Why  are  a  cf,  f  c  b,  right  angles  ? 
What  is  a  right  angle  ? 
Why  are  m  o  i,  j  o  p,  vertical  angles  ? 
What  are  vertical  angles  ? 
Why  is  m  o  i  an  exterior  angle  ? 
What  is  an  exterior  angle  ? 
Why  is  j  o  p  an  interior  angle  ? 
What  is  an  interior  angle  ? 
Why  are  m  o  i,  o  p  k,  opposite  angles  ? 
What  are  opposite  angles  ? 
Why  are  /  o  p,  o  p  fc,  alternate  angles  ? 
What  are  alternate  angles  ? 


38  FIRST  LESSONS  IN  GEOMETBY. 


LESSON     EIGHTEENTH. 

PROBLEMS. 

Draw  an  obtuse  angle  which  shall  be  only  a  little 

larger  than  a  right  angle. 
Draw  one  which  shall  be  much  greater  than  a  right 

angle. 
Draw  an   acute  angle  which  shall  be  only  a  little 

less  than  a  right  angle. 
Draw  orie  which  shall  be  much  less  than  a  right 

angle,. 
Draw  £,n  obtuse  angle  with  lines  about  one  inch 

long. 

Draw  an  acute  angle  with  sides  three  inches  long. 
Which  is  greater,  the  obtuse  angle,  or  the  acute 

angle  ? 

Draw  a  right  angle  with  lines  an  inch  long. 
Draw  one  with  lines  five  inches  long. 
Which  is  the  greater,  first  or  the  second  ? 


POLYGONS. 


39 


Diagram    16. 


40  FIRST  LESSONS  IN  GEOMETRY. 

LESSON    NINETEENTH. 

POLYGONS. 

Name  any  thing  besides  your  desk  thai  has  a  flat 

surface. 

A  flat  surface  is  called  a  plane. 
How  many  sides  has  the  plane  Fig.  A  ?    (DIAGRAM 

16.) 

It  is  called  a  triangle.     "  Tri "  means  "three." 
What  other  triangles  do  you  see. 
Triangles  are  sometimes  called  trigons. 

A.  triangle  is  a  plane  figure  having  tJiree  sides. 

How  many  sides  has  the  plane  figure  marked  B  ? 

How  many  angles  ? 
It  is  called  a  quadrarigle,  or  quadrilateral.    "  Quad  " 

denotes  "  four." 

What  other  quadrangles  do  you  see  ? 
Why  is  Fig.  B  a  quadrangle  ? 

A  quadrangle  is  a  plane  figure  Jiaving  four  sides. 

How  many  sides  has  the  Fig.  C  ? 
It  is  called  a  pentagon. 
What  other  pentagon  do  you  see  ? 
Why  is  Fig.  C  a  pentagon  ? 

A  pentagon  is  a  plane  figure  having  five  sides. 
In  like  manner,  — 

A  hexagon  is  a  plane  figure  having  six  sides. 
A  heptagon  is  a  plane  figure  Jiaving  seven  sides. 


POLYGONS.  41 

An  octagon  has  eight  sides. 

A  nonagon  has  nine  sides. 

A  defcagon  has  ten  sides. 

All  these  figures  are  called  polygons. 

"  Poly  "  means  "  many." 

What  do  you  call  a  polygon  of  three  sides  ?     Of 

four  sides  ?     Of  six  sides  ?  &c. 
If  the  length  of  each  side  of  triangle  A  is  one 

inch,  how  long  are  the  three  sides  together  ? 
The  sum  of  the  sides  of  a  polygon  is  its  perimeter. 
Which  of  the  triangles  has  unequal  sides  ?     Which 

has  equal  sides? 

The  latter  is  called  a  regular  polygon. 
Which  pentagon  has  one  side  longer  than  any  one 

of  its  other  sides  ? 
Which  has  its  sides  all  equal  to  each  other  ?     Are 

its  angles  also  equal  ? 

It  is  therefore  a  regular  polygon,  or  regular  pentagon. 
Name  a  hexagon  that  is  not  regular. 
Name  a  regular  hexagon. 

A  regular  octagon.     A  regular  heptagon. 

,/ 
A  polygon  is  a  plane  figure  bounded  \>y  straight  lines. 


42  FIRST  LESSONS   IN   GEOMETRY. 


LESSON    TWENTIETH. 

REVIEW. 

Name  all  the  triangles.     (DIAGRAM  16.) 

Why  is  Fig.  A  a  triangle  ? 

What  is  a  triangle  ? 

What  other  name  is  sometimes  given  to  triangles  ? 

Name  all  the  quadrilaterals. 

Why  is  Fig.  B  a  quadrilateral  ? 

What  is  a  quadrilateral,  or  quadrangle  ? 

Name    all    the    pentagons,   hexagons,   heptagons, 

octagons,  and  nonagons. 
Why  is  C  a  pentagon  ?     What  is  a  pentagon  ?     A 

hexagon  ?     A  heptagon  ?  &c. 
How  many  polygons  in  the  diagram  ? 
What  is  a  polygon  ? 
If  each    side    of   Fig.  B    is -one  inch,  how  many 

inches  are  there  in  its  perimeter  ? 
When  is  a  polygon  regular  ? 
Name  all  the  regular  polygons  in  diagram  16. 
Name  all  the  irregular  polygons. 


TRIANGLES. 


43 


a 


44  FIRST  LESSONS  IN   GEOMETRY. 


LESSON    TWENTY-FIRST. 

TRIANGLES. 
ACUTE-ANGLED  TRIANGLES. 

In  the  triangle  1,  what  kind  of  an  angle  is  1)  a  c  ? 

a  cl?  da?     (DIAGRAM  17.) 
Then  it  is  called  an  acute-angled  triangle. 

An  acute-angled  triangle  is  one  whose  angles  are  all 
acute. 

Read  three  other  acute-angled  triangles, 

OBTUSE-ANGLED  TRIANGLES. 

Iii  the  triangle  4,  what  kind  of  an  angle  is  /  Jc  m  ? 
Then  it  is  called  an  obtuse-angled  triangle. 

An  obtuse-angled  triangle  is  one  that  lias  one  obtuse 
angle. 

Name  two  others. 

RIGHT-ANGLED  TRIANGLES. 

In  ^he  triangle  3,  what  kind  of  an  angle  is  g  ij  ? 
Then  it  is  called  a  right-angled  triangle. 

A   right-angled   triangle   is   one   that   has    one    right 
angle. 

Name  three  other  right-angled  triangles. 
Upon  which  side  does  the  triangle  3  seem  to  stand  ? 
Then  ij  is  called  the  lose  of  the  triangle. 
What  letter  marks  the  vertex  of  the  angle  opposite 
the  base  ? 


TRIANGLES.  45 

Then  the  point  g  is  said  to  be  the  vertex  of  the 

triangle. 
If,  in  the  triangle  7,  we  consider  t  v  the  base,  what 

point  is  the  vertex  ? 
If  v  be  considered  the  vertex,  which  side  will  be 

the  base  ? 
In  the  triangle  3,  what  side  is  opposite  the  right 

angle  ? 
Then^y  is  called  the  hypothenuse  of  the  triangle. 

The  hypotJienuse  of  a  triangle  is  the  side  opposite  the 
right  angle. 

Eead  the  hypothenuse  of  each  of  the  triangles  5, 
6,  and  11. 

Either  side  about  the  right  angle  may  be  consid- 
ered the  base. 

Then  the  other  side  will  be  the  perpendicular. 

In  the  triangle  3,  if  ij  is  the  base,  which  side  is 
the  perpendicular? 

If  g  i  be  considered  the  base,  which  side  is  the  per- 
pendicular ? 

In  triangle  5,  if  n  o  is  the  base,  which  side  is  the 
perpendicular  ? 


46  FIRST  LESSONS  IN  GEOMETRY, 


LESSON    TWENTY-SECOND. 

REVIEW. 

Name  four  acute-angled  triangles.     (DIAGRAM  17.) 

Why  is  the  triangle  8  acute-angled  ? 

What  is  an  acute-angled  triangle  ? 

Name  three  obtuse-angled  triangles. 

Why  is  the  triangle  9  an  obtuse-angled  triangle  ? 

What  is  an  obtuse-angled  triangle  ? 

Name  four  right-angled  triangles. 

Why  is  the  triangle  6  a  right-angled  triangle  ? 

What  is  a  right-angled  triangle  ? 

In  the  triangle  6,  which  side  is  the  hypothenuse  ? 

Why? 

What  is  the  hypothenuse  ? 

What  two  sides  of  the  triangle  6  may  be  regarded 

as  the  base  ? 
If  q  r  be  considered  the  base,  what  do  you  call  the 

side  q  s  ? 
Read  the  hypothenuse  of  each  of  the  triangles  3 

5,  6,  and  11. 


TKIANGLES. 


47 


Diagram    18. 


48  FIKST  LESSONS  IN  GEOMETRY. 


LESSON    TWENTY-THIRD. 

TRIANGLES.     (Continued.) 
ISOSCELES  TRIANGLES. 

Of  the  triangle  1,  which  two  sides  are   equal  to 

each  other  ? 
Then  it  is  called  an  isosceles  triangle. 

An  isosceles  triangle  is  one  that  has  tivo  equal  sides. 

Name  eight  isosceles  triangles. 

Why  is  the  triangle  2  an  isosceles  triangle  ? 

What  kind  of  a  triangle  is  it  on   account  of  its 

angles  ? 

Then  it  is  an  acute-angled  isosceles  triangle. 
Name  four  acute-angled  isosceles  triangles. 
What  kind  of  a  triangle  is  Fig.  4  on  account  of  the 

angle  kj  I? 

What  kind  on  account  of  its  equal  sides  ? 
Then  it  is  called  an  obtuse-angled,  isoscetes  triangle. 
Name  one  other  obtuse-angled  isosceles  triangle. 
What  kind  of  a  triangle  is  Fig.  6  on  account  of  the 

angle  q  p  r  ? 

What  kind  on  account  of  its  equal  sides  ? 
Then  it  is  called  a  right-angled  isosceles  triangle. 
Name  one  other  right-angled  isosceles  triangle. 
Why  is  Fig.   12   a   right-angled    triangle?     Why 

isosceles  ? 


TKIANGLES.  49 

EQUILATERAL  TRIANGLES. 

Which  of  the  isosceles  triangles  has  all  its  three 

sides  equal  to  each  other? 
It  is  called  an  equilateral  triangle. 
"  Equi  "  means  "  equal."     "  Latus "  means  a  "  side." 

An  equilateral  triangle  is  one  that  has  its  three  sides 
equal  to  each  other. 

What  kind  of  a  triangle' is  Fig.  7  on  account  of  its 

three  equal  sides  ? 
What  kind  on  account  of  its  two  equal  sides  s  t,s  u. 

or  t  s,  t  u,  or  u  s,  u  t  ? 
Then  must  not  every  equilateral  triangle  be  also 

isosceles"? 
What  kind  of  a  triangle  is  Fig.  2  on  account  of  its 

equal  sides  d  e,  df? 
If  the  side  ef  is  longer  than  either  of  the  other 

two  sides,  is  it  an  equilateral  triangle  ? 
Then  is  every  isosceles  triangle  also  equilateral  ? 
Name  another  isosceles  triangle   that  is  not   equi- 
lateral. 

Name  one  that  is  equilateral. 
In  any  equilateral   triangle  the  three   angles  are 

equal  to  each  other. 
On  account  of  its  equal  angles,  it  is  also  called 

an  equiangular  triangle. 
What  is  Fig.  8  called  on  account  of  its  three  equal 

sides  ?     On  account  of  its  three  equal  angles  ? 
Every  equilateral  triangle  fe  also  equiangular. 
Every  equiangular  triangle  is  also  equilateral. 


50  FIRST  LESSONS  IN  GEOMETRY. 

Name  a  triangle  that  has  no  two  sides  equal  to 

each  other. 

It  is  called  a  scalene  triangle. 
What  kind  of  a  triangle  is  Fig.  5  on  account  of  its 

right  angle  ? 

What  kind  on  account  of  its  three  unequal  sides  ? 
Then  it  is  a  right-angled  scalene  triangle. 
What  name  can  you  give  Fig.  11  on  account  of  the 

angle  g  ef? 

On  account  of  its  three  unequal  sides  ? 
Then  what  may  it  be  called  ? 


TRIANGLES. 


51 


Diagram    19. 


52  FIRST   LESSONS   IN   GEOMETRY. 

LESSON    TWENTY-FOURTH. 

REVIEW. 

Name  eight  isosceles  triangles.     (DIAGRAM  19.) 

Why  is  Fig.  2  an  isosceles  triangle  ? 

What  is  an  isosceles  triangle  ? 

Name  two  right-angled  isosceles  triangles. 

Name  five  acute-angled  isosceles  triangles. 

Name  one  obtuse-angled  isosceles  triangle. 

Name  two  isosceles  triangles  that  are  also  equi- 
lateral. 

Are  all  isosceles  triangles  equilateral  ? 

Name  six  isosceles  triangles  that  are  not  equilateral. 

What  does  «  equi "  mean  ?     "  Latus  "  ? 

What  are  equilateral  triangles  called  on  account 
of  their  equal  angles  ? 

Are  all  equilateral  triangles  equiangular  ? 

Are  all  equiangular  triangles  equilateral  ? 

What  ar.e  equilateral  triangles  ? 

Name  four  scalene  triangles. 

Name  two  right-angled  scalene  triangles. 

Why  is  Fig.  3  a  right-angled  triangle?  Why 
scalene  ? 

What  is  a  scalene  triangle  ? 

Name  one  obtuse-angled  scalene  triangle. 

Name  one  acute-angled  scalene  triangle. 


TRIANGLES.  7\>  53 

/ 

PROBLEMS, 

From  the  same  point  draw  two  straight  lines  of  any 
length,  making  an  acute  angle  with  each  other. 

Make  them  equal  to  each  other  by  measuring. 

Join  their  ends. 

What  kind  of  a  triangle  is  it  on  account  of  its 
angles  ? 

On  account  of  its  two  equal  sides  ? 

Write  its  two  names  inside  of  it. 

Draw  an  isosceles  triangle  whose  equal  sides  shall 
each  be  less  than  the  third  side. 

Write  its  two  names  within  it. 

Draw  an  oblique  straight  line  twice  as  long  as  any 
short  measure  or  unit. 

At  one  end  draw  a  straight  line  perpendicular  to 
it,  and  three  times  as  long  as  the  same  measure. 

Connect  the  ends  of  the  two  lines  by  a  straight  line. 

What  kind  of  an  angle  is  that  opposite  the  last 
line  drawn  ? 

Are  any  two  of  its  sides  equal  ? 

Write  its  two  names  under  it. 

Draw  a  horizontal  straight  line  of  any  length. 

At  one  end  draw  a  vertical  line  of  equal  length. 

Complete  the  triangle,  and  write  two  names  inside. 

Draw  a  right-angled  triangle  whose  base  is  of  any 
length,  and  its  perpendicular  twice  as  long. 

Draw  a  right-angled  triangle  whose  base  is  three 
times  as  long  as  any  short  measure,  and  its  per- 
pendicular five  times  as  long  as  the  same  measure 
or  unit. 


FIRST  LESSONS   IN   GEOMETKY. 


Diagram   2O. 


QUADRILATERALS.  55 

QUADRILATERALS. 

How  many  sides  has  the  figure  a  b  d  c  ? 

What  is  it  called  on  account  of  the  .number  of  its 

sides  ? 
Name  three  other  quadrilaterals  whose  vertices  are 

marked. 

Name  seven  by  numbers. 
Quadrilaterals  are  sometimes  named  by  means  of 

two  opposite  vertices. 
The  quadrilateral  a  b  d  c,  or  c  d  b  a,  may  be  read 

a  d,  or  b  c,  or  c  £,  or  d  a. 
Name  the  quadrilateral,  g  hfe,  four  ways. 
How  many  angles  has  each  figure-? 
On  account  of  the  number  of  their  angles  they  are 

called  quadrangles. 
Has  the  quadrilateral  a  d  any  two  sides  parallel  to 

each  other  ? 
Then  it  is  called  a  trapezium. 

A  trapezium  is  a  quadrilateral  that  has  no  two  sides 
parallel. 

Name  two  other  trapeziums. 
Why  is  Fig.  7  a  trapezium  ? 

Has  the  quadrilateral  e  h  any  two  sides  parallel  ? 
Which  two  ?     Are  the  other  two  sides  parallel  ? 
It  is  called  a  "  trapezoid" 
"  Oid  "  means  like.    What  does  "  trapezoid  "  mean  ? 

A  trapezoid  is  a  quadrilateral  that  has  only  one  pair 
of  sides  parallel. 

Name  another  trapezoid. 


56  FIEST  LESSONS  IN  GEOMETRY. 

* 

Why  is  Fig.  6  a  trapezoid  ? 

How  many  pairs  of  parallel  sides  has  the  quadri- 
lateral il? 

Name  the  horizontal  parallels.    * 
Name  the  oblique  parallels. 
It  is  called  a  " parallelogram? 

A  parallelogram  is   a  quadrilateral  whose   opposite 
sides  are  parallel. 

Name  five  other  parallelograms. 

Why  is  Fig.  4  a  parallelogram  ? 

Why  is  not  Fig.  6  a  parallelogram  ? 

Why  is  not  e  li  a  parallelogram  ? 

What  two  names  may  you  giv6  to  Fig.  5? 

Why  is  it  a  quadrilateral  ?     Why  a  trapezium  ? 

What  two  names  may  we  give  to  Fig.  6  ? 

Why  is  it  a  quadrilateral  ?     Why  a  trapezoid  ? 

What  two  names  may  we  give  to  Fig.  3  ? 

Why  is  it  a  parallelogram  ?     Why  a  quadrilateral  ? 


QUADRILATEKALS.  57 


*  LESSON    TWENTY-FIFTH. 

REVIEW. 

How  many  quadrilaterals  in  the  diagram.  (DIA- 
GRAM 20.) 

Why  is  Fig.  a  d  a  quadrilateral  ? 

What  is  a  quadrilateral  ? 

On  account  of  the  number  of  its  angles,  what  may 
it  be  called? 

Name  all  the  quadrilaterals. 

Name  three  trapeziums. 

Why  is  Fig.  5  a  trapezium  ? 

What  is  a  trapezium  ? 

Name  two  trapezoids. 

*Why  is  Fig.  6  a  trapezoid  ? 

Name  its  parallel  sides. 

What  is  a  trapezoid  ? 

Name  six  parallelograms. 

Why  is  Fig.  4  a  parallelogram  ? 

Name  its  two  pairs  of  parallel  sides. 

What  is  a  parallelogram  ? 

What  two  names  can  you  give  to  Fig.  4  ? 

Why  the  first  ?     Why  the  second  ? 

What  two  names  may  be  given  to  Fig.  7  ? 

Why  the  first  ?     Why  the  second  ? 

What  two  to  Fig.  6  ? 

Why  the  first  ?     Why  the  second  ? 


58 


FIRST  LESSONS   IN   GEOMETRY. 


Quadrilateral. 
Parallelogram. 
Rhomboid. 


Quadrilateral. 
Parallelogram. 
Rectangle. 


Diagram   21. 


KINDS   OF   PARALLELOGRAMS.  59 


LESSON    TWENTY-SIXTH. 

KINDS    OF    PARALLELOGRAMS. 
BHOMBOIDS. 

How  many  quadrilaterals  in  the  diagram?     (DIA- 
GRAM 21.) 

How  many  parallelograms  ? 
Has  the  parallelogram  a  d  any  right  angle  ? 
It  is  called  a  "  rhomboid'' 

A  rhomboid  is  a  parallelogram  which  has  no  right 
angle. 

Name  five  other  rhomboids. 

What  three  names  may  be  given  to  Fig.  2  ? 

Why  is  it  a  quadrilateral  ? 

Why  a  parallelogram  ?     Why  a  rhomboid  ? 

BHOMBS. 

Are  the  four  sides  of  the  rhomboid  a  d  equal  to 

each  other? 
Are  the  four  sides  af  the  rhomboid  e  h  equal  to 

each  other  ? 
If  a  triangle  has  its  three  sides  equal  to  each  other, 

what  do  you  call  it  ? 
Then  when  a  rhomBoid  has  its  sides  equal  to  each 

other,  what  may  it  be  called  ? 
An  equilateral  rhomboid  is  called  a  rhombus. 

A  rhombus  is  an  equilateral  rhomboid. 

See  Note  D,  Appendix. 


60  FIRST  LESSONS   IN   GEOMETRY. 

Name  two  other  rhombuses,  or  rhombs. 
What  four  names  can  you  give  to  Fig.  e  U? 
Why    a    quadrilateral  ?      Why    a    parallelografti  ? 
Why  a  rhomboid  ?     Why  a  rhombus  ? 

RECTANGLES. 

Has  the  parallelogram  i  I  any  right  angles  ? 

How  many  ? 

It  is  called  a  "  rectangle." 

A  rectangle  is  a  right-angled  parallelogram. 

Name  four  other  rectangles. 
What  three  names  may  be  given  to  Fig.  i  I? 
Why   a   quadrilateral  ?      Why   a    parallelogram  ? 
Why  a  rectangle  ? 

SQUARES. 

Has  the  rectangle  i  I  its  four  sides  equal  ? 
Has  the  rectangle  m  p  its  four  sides  equal  ? 
It  is  called  a  "  square." 

A  square  is  an  equilateral  rectangle. 

Name  another  " square" 

What  four  names  may  be  given  to  Fig.  m  p  ? 
Why    a    quadrilateral  ?      Why    a    parallelogram  ? 
Why  a  rectangle  ?     Why  a  square  ? 


KINDS   OF   PARALLELOGRAMS.  61 


LESSON    TWENTY-SEVENTH. 

REVIEW. 

Name  six  rhomboids.     (DIAGRAM  21.) 

What  three  names  may  be  given  to  Fig.  3  ? 

Why    a    quadrilateral  ?      Why    a    parallelogram  ? 

Why  a  rhomboid  ? 
What  is  a  quadrilateral  ?  Parallelogram  ?  Ehom- 

boid? 

Name  three  rhombs. 
What  four  names  may  you  give  Fig.  5  ? 
Why    a    quadrilateral  ?      WJhy    a    parallelogram  ? 

Why  a  rhomboid  ?     Why  a  rhomb  ? 
What  is  a  rhomboid  ?     A  rhomb  ? 
Name  five  rectangles. 

What  three  names  may  be  given  to  Fig.  1  ? 
Why   a   quadrilateral  ?      Why   a    parallelogram  ? 

Why  a  rectangle  ? 
What  is  a  rectangle  ? 
Name. two  squares? 

By  what  four  names  may  Fig.  7  be  called  ? 
Why   by   the    first?     By    the    second?     By    the 

third  ?     By  the  fourth  ? 
What  is  a  square  ? 
What  is  a  rectangle  ? 
What  is  a  parallelogram  ? 
What  is  a  quadrilateral  ? 


62 


FIKST  LESSONS    EN   GEOMETBY. 


D 


E 


H 


Diagram    22. 


LESSON     TWENTY-EIGHTH. 

COMPAKISON   AND   CONTRAST. 
TBAPEZIUM    AND    TRAPEZOID. 

In  what  respect  are  Figs.  A  and  B  alike  ? 

On  this  account,  what  name  may  be  given  to  each  ? 

How  does  Fig.  B  differ  from  Fig.  A  ? 

What  particular  name  may  you  give  to  Fig.  B  ? 

What  one  to  Fig.  A  ? 


COMPARISON  AND   CONTRAST.  63 

KHOMBOID    AND    RECTANGLE. 

In  what  two  respects  are  Figs.  C  and  D  alike  ? 
On  account  of  the  number  of  their  sides,  what  may 

each  be  called  ? 
Because  their  opposite  sides  are  parallel,  what  may 

each  be  called  ? 

In  what  respect  do  they  differ  ? 
What  particular  name  may  be  given  to  Fig.  C  ? 
What  one  to  Fig.  D  ? 
What   three    names   may  you  give  to  the  figure 

with  right  angles  ? 
What  three  to  the  one  without  right  angles  ? 

RHOMBOID    AND    RHOMBUS. 

In  what  three  things  are  Figs.  E  and  F  alike  ? 
What  three  names  may  be  given  to  each  ? 
How  do  they  differ  from  each  other  ? 
What  particular  name  may  you  give  to  Fig.  F  ? 
What  four  names  has  Fig.  F  ? 

RECTANGLE    AND    SQUARE. 

In  what  three  things  are  Figs.  G  and  H  alike  ? 
On    account   of  the  number  of  their  sides,  what 

may  each  be  called  ? 
Because  their  opposite  sides  are  parallel,  what  may 

each  be  called  ? 
Because  they  have  right  angles,  what  may  they 

be  called? 
In  what  respect  is  Fig.  H  different  from  Fig.  G  ? 


64  FIRST  LESSONS   IN   GEOMETRY. 

On   this   account,  what  particular  name  may  be 

applied  to  Fig.  H  ? 

What  three  names  may  be  applied  to  Fig.  G  ? 
What  four  to  Fig.  H  ? 

RHOMBUS    AND    SQUARE. 

In  what  three  things  are  Figs.  F  and  H  alike  ? 
On  account  of  the  number  of  their  sides,  what 

name  may  be  given  to  each  ? 
Because    their   opposite    sides   are   parallel,  what 

name  may  be  given  to  each  ? 
Because  both  are  parallelograms,  and  both  have 

their  sides  equal,  what  name  may  be  given  to 

each? 

What  particular  name  has  Fig.  F  ? 
What  particular  name  has  Fig.  H  ? 
What  four  names  may  be  given  to  Fig.  F  ? 
What  four  to  Fig.  H? 


COMPARISON  AND   CONTRAST.  65 

LESSON    TWENTY-NINTH. 

REVIEW. 

What  two  names  may  be  given  to  Fig.  A.     (DIA- 
GRAM 22.) 
To  Fig.  B  ? 

In  what  are  they  alike  ? 
In  what  do  they  differ  ? 

By  what  three  names  may  Fig.  C  be  called  ? 
By  what  three  names  may  Fig.  D  be  called  ? 
In  what  two  things  are  they  alike  ? 
In  what  one  thing  do  they  differ  ? 
What  particular  name  has  C  ?     What  one  has  D  ? 
What  three  names  may  be  applied  to  Fig.  E  ? 
What  four  to  Fig.  F  ? 
What  property  has  F  that  E  has  not  ? 
What  particular  name  has  it  on  that  account? 
What  three  names  has  Fig.  G  ? 
What  four  has  Fig.  H  ? 
What  property  has  Fig.  H  that  G  has  not  ? 
What  particular  name  has  it  in  consequence  ? 
What  four  names  may  you  give  to  Fig.  F  ? 
What  four  to  Fig.  II  ?  * 

What  three  names  may  be  applied  to  either  ? 
In  what  three  things  are  they  alike  ? 
In  what  respect  do  they  differ  ? 
What  particular  name  has  Fig.  F  ? 
What  particular  name  has  Fig.  H  ? 


G6 


FIRST  LESSONS   IN   GEOMETRY. 


Diagram    23. 

LESSON    THIRTIETH. 

MEASUREMENT   OF   SURFACES. 

In  Fig.  1  (DIAGRAM  23)  call  the  line  a  b  a  unit. 

Rectangle  1  is  how  many  units  long  ? 

How  many  high  ? 

Because  its  sides  are  equal,  what  is  it  called  ? 

Rectangle  2  is  how  many  units  long  ? 


MEASUREMENT  OF  SURFACES.          67 

*How  many  high  or  wide? 
How  many  squares  does  it  contain  ? 
Rectangle  3  is  how  many  units  long  ? 
How  many  wide  ? 

How  many  squares  does  it  contain  ? 
If  it  were  four  units  long  and  one  wide,  how  many 

squares  would  it  contain  ? 

If  it  were  five  long  and  one  wide  ?     Six  long?  &c. 
Rectangle  4  is  how  many  units  long  ? 
How  many  wide  ? 

How  many  squares  does  it  contain  ? 
How  many  squares  in  that  part  which  is  two  units 

long,  m  n,  and  one  unit  wide,  m  I? 
On  account  of  the  second  unit  in  width,  /  k,  how 

many  times  two  squares  are  there  ? 
If  the  width  were  one  unit  more,  how  many  times 

two  squares  would  there  be  ? 
Rectangle  5  is  how  many  units  long  ? 
How  many  units  wide  or  high  ? 
How  many  squares  does  it  contain? 
How  many  squares   in    that  part  which   is  three 

*  units  long,  op,  and  one  unit  wide,  ot? 
The    second   unit  in  width,  t  q,  gives  how  many 

more  squares  ?     How  many  times  three  squares  ? 
If  another  unit  were    added    to  the   width,  how 

many  more  squares  would  be  made  ? 
How  many  times  three  squares  ? 
If  it  were  four  units  wide,  how  many  times  three 

squares  would  there  be  ? 
Rectangle  6  is  how  many  units  long? 


68  FIRST   LESSONS   IN   GEOMETRY. 

How  many  units  high  or  wide  ? 

How  many  squares  in  that  part  which  is  four  units 

long  and  one  high  ? 
How  many  times  four  squares  in  that  part  which 

is  four  long  and  two  high  ? 
How  many  times  four  squares  when  it  is  four  long 

and  three  high  ? 
If   another  unit  were    added  to  the  height,  how 

many  more  squares  would  be  added  ? 
How  many  times  four  squares  would  there  be  ? 
If  a  rectangle  were  five   units  long  and  one  unit 

wide,  how  many  square  units  would  it  contain  ? 
If  it  were  two  units  wide,  how  many  times  live 

square  units  would  it  contain  ? 
If  it  were  three  units  wide  ?     Four  ?  &c. 
If  your  ruler  is  ten  inches  long  and  only  one  inch 

wide,  how  many  square  inches  are  there  in  it  ? 
If  it  were  two  inches  wide,  how  many  times  ten 

square  inches  would  it  contain  ? 
If  your  arithmetic-cover  is  seven  inches  long  and 

five  inches  wide,  how  many  square  inches  are 

there  in  it  ?  ** 

If  a  wall  of  this   room  is  twenty  feet  long,  how 

many  square  feet  are  there  in  that  part  which  i.; 

one  foot  high  ?    Two  high  ?    Three  high  ?    Four 

high? 
If  the   same  wall  is  sixteen  feet  high,  how  many 

square  feet  in  it  ? 
Fig.  5  has  how  many  times  three  squares  ? 


MEASUREMENT  OF  SURFACES.          69 

Fig.  7  has  how  many  times  two  squares  ? 
Which  has  the  greater  number  of  squares  ? 
What  difference  is  there  between  two  times  three 
squares  and  three  times  two  squares  ? 


LESSON    THIRTY-FIRST. 

REVIEW. 

Draw  a  rectangle   of  any  width  whose  length  is 

three  times  the  width. 
How  many  squares  has  it  if  the  width  be  taken  as 

the  unit  ? 

Make  it  twice  as  wide  as  before. 
How  many  squares  has  it  now  ? 
What  two   numbers  multiplied  together  will  give 

the  number  of  squares  ? 
Make  it  three  times  as  wide. 
How  many  squares  has  it  now  ? 
What  two  numbers  multiplied  together  will  give 

the  number  of  squares  ? 
The  cover  of  a  geography  is  one  foot  long  and  one 

foot  wide,  how  many  square  feet  in  it  ? 
How  many  inches  long  is  the  same  cover  ?     How 

many  wide  ? 

How  many  square  inches  does  it  contain  ? 
How  many  square  inches  are  equal  to  one  square 

foot? 


70  FIHST   LESSONS    IN   GEOMETRY. 

A  table  is  one  yard  long  and  one  yard  wide,  how 

many  square  yards  in  it  ? 
How  many  feet  long  is  the  same  table  ? 
How  many  feet  wide  ? 
How  many  square  feet  does  it  contain  ? 
One  square  yard  equals  how  many  square  feet? 
Draw  a  square  whose  side  is  a  unit  of  any  length. 
Draw  another  whose  side  is  two  units  of  the  same 

length. 
The  second  square  is  how  many  times  as  large  as 

the  first  one  ? 

How  many  squares  in  half  the  second  square  ? 
Which  is  greater,  two  square  inches,  or  two  inches 

square  ? 
Two  inches  square  is  how  many  times  two  square 

inches  ? 

Draw  a  square  whose  side  is  three  inches. 
How  many  square  inches  does  it  contain  ? 
How  many  times  as  many  squares  as  the  square 

of  one  inch  ? 

How  many  square  inches  in  the  bottom  row  ? 
How  many  in  all  ? 
Which  is  greater,  three  inches  square,  or  three 

square  inches  ? 
Three   inches   square   is   how  many  times    three 

square  inches  ? 


MEASUREMENT   OF   SURFACES.  71 


PROBLEMS. 

An  equilateral  triangle  has  each  of  its  sides  one 

inch  long,  what  is  its  perimeter  ? 
If  each  side  were  two  inches  long,  what  would  be 

its  perimeter  ? 
An  isosceles  triangle  has  its  two  equal  sides  each 

three  inches  long,  and  its  third  side  five  inches 

long,  what  is  its  perimeter  ? 
A  right-angled  isosceles  triangle  has  its  base  five 

inches,  and  its  hypothenuse  seven  inches  long, 

what  is  its  perimeter  ? 
A  square  geography-cover  is  nine  inches  long  on 

one  side,  how  long  all  round  ? 
How  many  square  inches  in  it  ? 
A "  slate  is  sixteen   inches  long  and  twelve  wide, 

how  many  inches  all  round  it  ? 
A  rectangle  is  five  inches  long  and  three  wide,  how 

long  all  round  ? 

How  many  square  inches  in  it  ? 
A  slate  is  one   foot  long  and  eight  inches  wide, 

what  is  its  perimeter  ? 
A  room  is  twenty-four  feet  long  and  twenty-one 

feet  wide,  how  many  feet  all  round  it  ? 
How  many  square  feet  in  the  floor  ? 
How   many  pieces   of  paper  each  a   foot   square 

would  exactly 'cover  it? 
A   yard    of   carpet   is   two  feet  wide,  how  many 

square  feet  in  it  ? 


72  FIRST  LESSONS   IN   GEOMETRY. 

Charles  and  Henry  start  from  the  same  place,  and 
walk  in  opposite  directions ;  Charles  goes  twenty 
yards,  and  Henry  fifteen,  how  many  yards  apart 
are  they  ? 

If  they  start  from  opposite  ends  of  a  straight  walk 
twenty-five  feet  long,  and  walk  towards  each 
other,  how  many  feet  will  Charles  have  to  walk 
to  meet  Henry  who  has  walked  fifteen  feet  ? 

A  lot  is  forty  rods  long  and  thirty  wide,  how  long 
must  the  fence  be  ? 

What  length  of  fence  will  divide  it  into  four 
equal  parts  ? 


THE   CIRCLE  AND   ITS   LINES. 


73 


LESSON    THIRTY-SECOND. 

THE    CIRCLE    AND    ITS    LINES. 

If  the  straight  line  c  a  were  a  string  made  fast  at 
c,  with  a  sharp  pencil-point  at  the  other  end  a, 
and  the  pencil-point  were  moved  towards  d,  what 
line  would  be  drawn  ? 

What  kind  of  a  line  would  it  be  ? 

If  the  pencil-point  continued  to  move  in  the  same 
direction  until  it  returned  to  the  starting-point 
a,  what  curved  line  would  be  drawn,  naming  it 
by  all  the  points  in  it  which  are  marked  ? 


74  FIRST  LESSONS  IN  GEOMETRY. 

The  plane  figure  bounded  by  this  curve  is  called  a 

"circle." 
What  point  is  at  the  centre  of  this  figure  ? 

A  circle  is  a  plane  figure  bounded  by  a  curved  line, 
all  points  of  which  are  equally  distant  from  the 
centre. 

The  curved  line  is  called  a  " circumference" 

The  circumference  of  a  circle  is  the  curve  which 
bounds  it. 

Name   a  straight  line  that  joins  two  points  in  the 

circumference. 
It  is  called  a  "  chord." 

A  chord  is  a  straight  line  that  joins  two  points  of  a 
circumference. 

Head  six  chords  in  the  diagram. 

Which  two  of  these  chords  pass  through  the  centre  ? 

They  are  called  " diameters" 

A  diameter  is  a  chord  that  passes  through  the  centre. 

Name  a  line  that  joins  the  centre  with  a  point  of 

the  circumference. 
It  is  called  a  "radius"  —  (Plural,  radii.} 

A  radius  is  a  straight  line  that  joins  the  centre  to  a 
point  of  the  circumference. 

Read  five  radii. 

Which  is  farther  from  the  centre,  the  point  a  or 

the  point  d? 
Can  the  radius  c  d  be  greater  than  the  radius  c  a  ? 

Or  greater  than  c  v,  or  c  of 


-  •  THE   CIRCLE   AND   ITS   LINES.  75 

Then  all  radii  of  the  same  circle  are  equal  to  each  other. 
What  do  we  call  the  lines  o  d,  c  d,  c  o  ? 
What  part  of  the  diameter  o  d  is  the  radius  o  c  ? 
Name  a  chord  that  is  produced  without  the  circle. 
It  is  called  a  "  secant." 

A  secant  is  a  chord  produced. 

Name  two  secants. 

If  the   chord  d  i  were  made  a  secant,   would  it 

become  longer  or  shorter  ? 
In  how  many  points  does  the  straight  line  I  m 

touch  the  circumference  ? 
It  is  called  a  "  tangent." 

A  tangent  is  a  straight  line  that  touches  a  circumfer- 
ence in  only  one  point. 

Name  three  tangents. 


LESSON    THIRTY-THIRD. 

REVIEW. 

Read  six  chords.     (DIAGRAM  24.) 

Why  is  i  d  a,  chord  ? 

What  is  a  chord  ? 

Name  two  diameters. 

Why  is  aj  a  diameter? 

What  is  a  diameter? 

Is  every  chord  a  diameter  ? 


76  FIRST  LESSONS  IN   GEOMETRY. 

Is  every  diameter  a  chord  ? 
Name  five  radii. 
Why  is  c  a  a  radius  ? 
What  is  a  radius  ? 

A  diameter  is  equal  to  how  many  radii  ? 
Are  all  radii  equal  to  each  other  ? 
Are  all  chords  equal  to  each  other  ? 
Are  all  diameters  equal  to  each  other?. 
Name  two  secants. 
Why  is  either  one  a  secant? 
What  is  a  sgcant  ? 
Name  three  tangents. 
Why  is  a  b  a  tangent  ? 
What  is  a  tangent  ? 

Is  a  tangent  inside  of  a  circle  or  outside  of  it  ? 
Is  a  chord  inside  or  outside  of  a  circle  ? 
Is  a  secant  within  or  without  a  circle  ? 
tf  the  radius  is  three  inches,  how  long  is  the  diam- 
eter? 


ARCS  AND   DEGEEES. 


77 


Diagram    25. 


78  FIRST  LESSONS   IN   GEOMETRY. 

LESSON     THIRTY-FOURTH. 

ARCS   AND   DEGREES. 

What  small  part  of  the  circumference  of  circle  1 

(DIAGRAM  25)  is  marked? 
It  is  called  an  "  arc.'9 

An  arc  is  any  part  of  a  circumference. 

Read  five  arcs  that  are  marked. 

Which  is  longer,  the  arc  e  d,  or  the  arc  ef?  b  d,  or 

1)  d  e  ?  a  b  d,  or  a  b  d  e  ? 

Name  an  arc  which  is  half  of  the  circumference. 
It  is  called  a  "  semi-circumference? 
"Semi"  means  "half." 

A  semi-circumference  is  half  of  a  circumference. 

Read  three  arcs,  each  of  which  is  one-fourth  of  the 

circumference. 
If  the  whole  circumference  were  divided  into  three 

hundred  and  sixty  equal  arcs,  would  each  arc  be 

large  or  small  ? 
Each  of  these  arcs  would  be  called  a  "degree? 

[Degrees  are  marked  (°).] 

A  degree  of  a  circumference  is  a  three  hundred  and 
sixtieth  part  of  it. 

How  many  degrees  in  a  semi-circumference  ? 

How  many  degrees  in  one-fourth  of  a  circum- 
ference ? 

If  a  fourth  of  a  circumference  were  divided  into 
three  equal  parts,  how  many  degrees  would 
there  be  in  each  part  ? 


AKCS   AND  DEGREES.  79 

Into  how  many  parts  would  each  third  of  a  quarter 

have  to  be  again  divided  to  make  single  degrees  ? 
Is   an  arc  of  ninety-one    degrees   greater  or  less 

than  one-fourth  of  a  circumference  ? 
Is  an  arc  of  a  hundred  and  seventy-nine  degrees 

greater  or  less  than  a  semi-circumference  ? 
Can  there  be  more   than  three  hundred  and  sixty 

degrees  in  a  circumference  ? 
If  the  circumference  of  circle  1  were  divided  into 

degrees,  each  degree  would  be  so  small  an  arc 

that  it  would  look  like  a  dot. 
If  a  degree  were   divided  into  sixty  equal  parts, 

each  part  would  be  called  a  minute. 
If  a  minute  were  divided  into  sixty  equal  parts, 

each  part  would  be  called  a  second. 
How  many  degrees  in  the  large  circle  of  Fig.  2  ? 
How  many  in  the  smaller  one  ? 
Has  a  large  circle  any  more  degrees  than  a  small 

circle  ? 

In  the  large  circle  how  many  degrees  from  a  to  b? 
In  the  small  circle  how  many  from  a  to  b? 
Which  is  greater,  an  arc  of  ninety  degrees  of  the 

large   circle,  or  one  of  ninety  degrees   of  the 

small  one  ? 
Which  is  greater,  an  arc  of  a  degree  of  the  large 

circle,  or  one  of  a  degree  of  the  small  one  ? 
The  angle  a  o  b  has  its  vertex  at  what  part  of  the 

larger  circle  ? 

At  what  part  of  the  smaller  circle  ? 
On  how  many  degrees  of  the  larger  circle  does  the 

angle  stand  ? 


80  FIRST  LESSONS  IN  GEOMETRY. 

On  how  many  degrees  of  the  smaller  circle  does  it 

stand  ? 

Then- it  is  said  to  be  an  angle  of  90°. 
If  the  angle  a  of  is  an  angle  of  30°,  how  many 

degrees  must  there  be  in  the  arc  af? 
If  the  arc  /  e  is  an  arc  of  60°,  what  is  the  size  of 

the  angle  f  o  e? 
An  angle  of  10°  stands  upon  an  arc  of  how  many 

degrees?     Of  8°?     Of  1°? 
The  angle  a  o  b  is  what  kind  of  an  angle  ? 
Upon  how  many  degrees  does  it  stand  ? 
Then   a   right  angle  is  an  angle    of  how  many 

degrees  ? 
If  an  angle  stand  upon  less  than  90°,  what  kind  of 

an  angle  is  it  ? 
If  an  angle  stand  upon  more  than  90°,  what  kind 

of  an  angle  is  it  ? 
Can  an  angle  have  as  many  degrees  as  a  hundred 

and  eighty  ? 


ARCS  AND  DEGREES.  81 


LESSON    THIRTY-FIFTH. 

REVIEW. 

Read  nine  arcs  whose  ends  are  marked.     (DIAGRAM 

26.) 
Read  three  arcs  each  of  which  is  one-fourth  of  a 

circumference. 

Read  two  arcs  each  of  which  is  one-half  of  a  cir- 
cumference. 
Why  is  e  g  an  arc  ? 
What  is  an  arc  ? 

How  many  degrees  in  the  arc/^/     In  e  h? 
If  the  arc  /  h  wrere  divided  into  three  equal  parts, 

how  many  degrees  would  there  be  in  each  ? 
How  marny  degrees  in  a  circumference  ? 
In  a  semi-circumference  ? 
How  many  more  degrees  in  a  large  circumference 

than  in  a  small  one  ? 
If  the  arc  if  is  40°,  what  is  the  size  of  the  angle 

/  o  i  ? 
If  the  angle  /  o  g  is  an  angle  of  130°,  what  is  the 

size  of  the  arc/  i  kg? 
How  many  degrees  in  each  of  the  adjacent  angles 

/  o  h,  h  o  e  ? 
When  two  adjacent  angles  are  equal  to  each  other, 

wrhat  is  each  called  ? 
How  many  degrees  in  a  right  angle  ? 


82 


FIRST  LESSONS   IN   GEOMETRY. 


Diagram   26. 


LESSON    THIRTY-SIXTH. 

PARTS   OF   THE   CIRCLE. 

The  part  of  the  circle  bounded  by  the  chord  a  b 

and  the  arc  a  b  is  called  a  segment. 
Read  three  segments,  each  less  than  half  a  circle, 

thus,  — •  the  segment  bounded  by  the  chord  a  d 

and  the  arc  a  b  d. 

A  segment  is  a  part  of  a  circle  bounded  by  an  arc 
and  a  chord. 

Read  two  segments  that  are  each  half  a  circle. 
What  is  the  chord  called  ? 
What  is  the  arc  called  ? 


PAKTS   OF  THE   CIRCLE.  83 

A  segment  bounded  by  a  diameter  and  a  semi- 
circumference  is  a  "semicircle" 

A  semicircle  is  half  a  circle. 

Read  four  segments  each  larger  than  a  semicircle. 
The   part  of  the  circle  between  the  two  radii  of, 

o  i,  and  the  arc/ i,  is  called  a  " sector" 
Read  four  sectors  each  less  than  one-fourth  of  a 

circle.* 

A  sector  is  a  part  of  a  circle  bounded  l>y  two  radii 
and  an  arc. 

What  part  of  the  whole  circle  is  the  sector/ oh? 
It  is  called  a  "  quadrant" 

A  quadrant  is  a  sector  which  is  one-fourth  of  a  circle. 

Read  a  sector  which  is  greater  than  a  quadrant. 

If  the  chord  efbe  regarded  a  diameter,  what  do 
you  call  the  semicircle  below  it  ? 

If  it  be  regarded  as  two  radii,  what  is  the  semi- 
circle called  ? 

Then  a  semicircle  is  both  a  segment  and  a  sector. 

*  Thus,  a  sector  bounded  by  the  two  radii  o  g,  o  h,  and  the  arc  g  h. 


84  FIBST  LESSONS  IN  GEOMETRY. 


LESSON    THIRTY-SEVENTH. 

REVIEW. 

Name  ten  segments.     (DIAGRAM  26.) 

What  is  a  segment  ? 

Of  the  segments  named,  which  are  less  than  a 

semicircle  ? 
Which  are  greater  ? 
Which  two  are  semicircles  ? 
Which  two  are  on  the  chord  aff 
Name  nine  sectors. 
Why  is  g  o  i  a  sector  ? 
What  is  a  sector  ? 
Which  four  of  the  sectors  named  are  each  less  than 

a  quadrant  ? 

Which  three  are  quadrants  ? 
Which  two  are  greater  than  a  quadrant  ? 
What  part  of  the  circle  is  both  a  segment  and  a 

sector  ? 

How  many  quadrants  in  a  circle  ? 
How  many  semicircles  ? 


PAET  SECOND, 


AXIOMS  AND  THEOREMS. 


AXIOMS   ILLUSTRATED. 

AXIOM  1. 

The  triangle  A  is  equal 

to  the  triangle  C. 
The   triangle    B    is   also 

equal  to  the  triangle  C. 
What  do  you  think  of  the  two 

triangles  A  and  B  ?     Why  ? 

If  two  things  are  separately  equal  to  the  same  thing, 
they  are  equal  to  each  other. 


AXIOM  2. 

The  square  A  is  equal  to 
the  square  B. 

To  the  rectangle  C  add  the 
square  A,  and  we  have  an 
L  pointing  in  what  direc- 
tion ? 

To  the  same  rectangle  C 
add  the  square  B?  and  we 


have  an  L  pointing  in  what  direction  ? 


86 


FIRST  LESSONS  IN  GEOMETRY. 


Which  is  larger,  the  L  pointing  to  the  left,  or  that 

pointing  to  the  right  ? 

To  what  same  thing  did  you  add  two  equals  ? 
What  two  equals  did  you  add  to  it  ? 
What  was  the  first  surn  ? 
The  second  ? 
What  do  you  think  of  the  two  sums  ? 

If  equals  be  added  to  the  same  thing,  the  sums  will 
be  equal. 

AXIOM  3. 


The   square  A  is  equal  to 

the  square  B. 
From  the  inverted  T  take 

away  the  square  A,  and 

we  have  an  L  pointing  in 

what  direction  ? 
From  the  same  Fig.  T  take 

away  the  square  B,  and 

we  have  an  L  pointing  in  what  direction  ? 
Which  is  larger,  the  L  pointing  to  the  right,  or 

that  pointing  to  the  left  ? 
What  two  equal  things  did  we  take  away  from  the 

same  thing  ? 

From  what  same  thing  did  we  take  them  away  ? 
What  did  we  find  true  of  the  two  remainders  ? 


If  equals  be  taken  from  the  same  thing 9  the  remain- 
ders will  be  equal. 


AXIOMS   ILLUSTRATED. 


87 


AXIOM  4. 

The  rectangle  1  2  is  equal 

to  the  rectangle  1  3. 
From    the    rectangle    1    2 

take  away  the  square  A, 

and   what   rectangle   re- 
mains ? 
From    the    rectangle    1   3 

take    away    the    same 

square  A,  and  what  rectangle  remains  ? 
Which  is  greater,  the  rectangle  B,  or  the  rectangle 

C? 

What  same  thing  did  we  take  away  from  equals  ? 
From  what  did  we  first  take  it  ? 
What  remained  ? 
From  what  did  we  next  take  it  ? 
What  remained  ? 
What  did  we  find  true  of  the  two  remainders  ? 

If  the  same  thing  be  taken  from  equate,  the  remain- 
ders will  be  equal. 

AXIOM  5. 

If  equals  be  added  to  equals,  the  sums  will  be  equal. 

AXIOM  6. 

If  equals  be  subtracted  from  equals,  the  remainders 
will  be  equal. 

AXIOM  7. 

If  the  halves  of  two  things  are  equal,  the  wholes  will 
be  equal. 


88  FIRST  LESSONS  IN  GEOMETRY. 

AXIOM  8. 

Every  iitfiole  is  equal  to  the  sum  of  all  its  parts. 

AXIOM  9. 

From  one  point  to  another  only  one  straight  line  can 
be  draivn. 

AXIOM  10. 

A  straight  line  is  the  shortest  distance   between  two 
points. 

AXIOM  11. 

If  two  things  coincide  throughout  their  whole  extent, 
they  are  equal. 


THEOREMS    ILLUSTRATED. 


Diagram    29. 


DEVELOPMENT    LESSON. 


Do  the  angles  Blue,  Red,  take  up  all  the  space  on 

the  line  a  b  ? 
Do  the  angles  Blue,  Yellow,  Red,  take  up  all  the 

space  on  the  line  ? 


THEOREMS   ILLUSTRATED.  89 

Do  the  angles  Blue,  Yellow,  Green,  Red,  take  up 

all  the  space  on  the  line  ? 
Is  there  room  between  any  two  of  the   angles  to 

put  in  another  angle  ? 
Then  are  not  the  angles  Blue,  Yellow,  Green,  Red, 

equal  to  all  the  space  on  the  line  a  I  ? 

NOTE.  —  The  word  space,  as  here  used,  means  angular  space;  and 
it  is  indispensable  that  the  teacher  impress  this  fact  upon  the  learner. 
By  means  of  former  lessons,  the  pupil  has  learned  positively,  that 
an  angle  is  the  difference  between  the  directions  of  two  lines ;  and, 
impliedly,  that  the  included  space  has  nothing  to  do  with  the  size  of 
the  angle.  There  cannot,  therefore,  be  much  danger  that  the  pupil 
will  imbibe  any  erroneous  notion  from  this  style  of  expression,  which 
is  very  much  more  simple  than  to  say  that  the  difference  of  direction 
of  two  given  lines  is  equal  to  the  difference  of  direction  of  two  other 
given  lines,  which  style  •will  be  used  somewhat  later  in  these  lessons. 


90  FIRST  LESSONS  IN  GEOMETRY. 


PROPOSITION    I.     THEOREM. 

DEVELOPMENT  LESSON. 

Are  the  adjacent  angles  Green,  Red,  equal  to  all 

the  angular  space  on  the  line  a  ~b? 
Place  a  paper  square  corner  or  right  angle  on  the 

line  a  b  at  the  left  of  c  d  with  its  vertex  at  c. 
It  will  cover  all  the  angle  Green  and  part  of  the 

angle  Red  up  to  the  line  c  d. 
Now  place  another  square  corner  on  the  line  a  b  to 

the  right  of  the  line   c  d,  and  with  its  vertex  at 

the  point  c. 
It  will  cover  the  remaining  part  of  the  angle  Red, 

and  two  edges  of  the  square  corners  will  meet 

along  the  line  c  d. 
Are  the  two  right  angles  equal  to  all  the  angular 

space  on  the  line  ab? 
Then  if  the  two  adjacent  angles  Green,  Red,  are 

equal  to  all  the  angular  space  on  the  line  a  b, 

and   the  tvyo   right  angles  are  also  equal  to  the 


THEOREMS   ILLUSTIiATED.  91 

same  space,  what  do  you  infer  concerning  the 
adjacent  angles  and  the  two  right  angles  ? 

What  axiom  do  you  apply  when  you  say  that 
the  adjacent  angles  are  equal  to  the  two  right 
angles  ? 

To  what  same  thing  did  you  find  two  things  sepa- 
rately equal  ? 

What  did  you  first  see  equal  to  it  ? 

What  did  you  next  see  equal  to  it  ? 

Then  what  did  you  find  true  ? 

If  the  angle  Red  were  smaller,  and  the  angle 
Green  larger,  would  the  adjacent,  angles  still  be 
equal  to  two  right  angles  ? 

Then,- 

Any  two  adjacent  angles  are  equal  to  two  r\ght  angles. 

If  we  draw  the  straight  line  c  d  where  the  edges 
of  the  square  corners  come  together,  what  kind 
of  angles  will  a  c  d,  d  c  b,  be  ? 

See  now  if  you  can  understand  the  following 
demonstration ;  — 

DEMONSTRATION. 

We  wish  to  prove  that 

Any  two  adjacent  angles  are  equal  to  two  right  angles. 

Let  the  two  straight  lines  a  b,  m  n,  intersect  each 

other  in  the  point  c.     (DIAGRAM  30.) 
Then  will  any  two  adjacent  angles,  as  Green,  Red, 

be  equal  to  two  right  angles? 


92  FIRST  LESSONS  IN  GEOMETEY. 

For,  from  the  point  <?,  draw  the  straight  line  c  d  so 
as  to  make  the  angles  a  c  d,  d  c  b,  right  angles. 

The  adjacent  angles  Green,  Red,  are   equal  to  all 

•    the  angular  space  on  the  line  a  b. 

The  right  angles  a  c  d,  d  c  b,  are  also  equal  to  all 
the  angular  space  on  the  line  a  b. 

Therefore  the  adjacent  angles  Green,  Red,  are 
equal  to  two  right  angles. 

TEST   QUESTIONS. 

To  what  same  thing  did  you  find  two  things  equal  ? 
What  did  you  first  see  equal  to  it  ? 
What  did  you  next  see  equal  to  it  ? 
Then  what  new  thing  did  you  find  true  ? 
What  axiom  did  you  make  use  of? 


THEOREMS  ILLUSTRATED. 


93 


Diagram   31. 


TEST    LESSON. 

By  means  of  Fig.  A,  — 

1.  Prove  that  the  adjacent  angles  Green,  Bed,  are 
equal  to  two  right  angles. 

2.  Prove  that  the  adjacent  angles   Blue,  Yellow, 
are  equal  to  two  right  angles. 

By  means  of  Fig.  B, — 

3.  Prove  that  the  adjacent  angles  Green,  Red,  are 
equal  to  two  right  angles. 

4.  Prove   that  the  adjacent  angles  Yellow,  Blue, 
are  equal  to  two  right  angles. 


94  FIRST  LESSONS  IN  GEOMETRY. 

By  means  of  Fig.  C, — 

5.  Prove  that  the  adjacent  angles  Red,  Blue,  are 
equal  to  two  right  angles. 

6.  Prove  that  the  adjacent  angles  Green,  Yellow, 
are  equal  to  two  right  angles. 

7.  Give  the  preceding  demonstrations   again,  but 
name  the  angles  by  their  letters  instead  of  by 
their  colors. 


THEOREMS  ILLUSTRATED. 


95 


96  FIKST  LESSONS   IN   GEOMETKY. 


TEST    LESSON. 

By  means  of  Fig.  A  prove,  — 

1.  That  the  adjacent  angles  a  c  m,  m  c  b,  are  equal 
to  two  right  angles. 

2.  That  the  adjacent  angles  a  c  n,  n  c  b,  are  equal 
to  two  right  angles. 

By  means  of  Fig.  B  prove,  — 

3.  That  the  adjacent  angles  a  c  n,  n  c  b,  are  equal 
to  two  right  angles. 

4.  That  the  adjacent  angles  a  c  m,  m  c  b,  are  equal 
to  two  right  angles. 

By  means  of  Fig.  C  prove,  — 

5.  That  the  adjacent  angles  a  c  m,  m  c  b,  are  equal 
to  two  right  angles. 

6.  That  the  adjacent  angles  a  c  n,  n  c  b,  are  equal 
to  two  right  angles. 

By  means  of  Fig*  D  prove,  — 

7.  That  the  adjacent  angles  a  c  n,  n  c  b,  are  equal 
to  two  right  angles. 

8.  That  the  adjacent  angles  b  c  m,  m  c  a,  are  equal 
to  two  right  angles* 


THEOREMS   ILLUSTRATED. 


97 


Diagram    33. 

PROPOSITION    II.      THEOREM. 

DEVELOPMENT   LESSON. 

What  kind  of  angles  are  P  and  S  ? 

How  do  the  adjacent  angles  Yellow,  Blue,  compare 
with  the  right  angles  P,  S  ? 

How  do  the  adjacent  angles  Blue,  Red,  compare 
with  the  tvyo  right  angles  ? 

Then  if  the  adjacent  angles  Yellow,  Blue,  are  equal 
to  two  right  angles,  and  the  adjacent  angles 
Blue,  Red,  are  also  equal  to  two  right  angles, 
what  do  you  think  of  the  two  pairs  of  adjacent 
angles,  Yellow,  Blue,  and  Blue,  Red  ? 

If,  from  the  adjacent  angles  Yellow,  Blue,  we  take 
away  the  angle  Blue,  what  remains  ? 

If,  from  the  adjacent  angles  Blue,  Red,  we  take 
away  the  same  angle  Blue,  what  remains  ? 


98  FIRST  LESSONS  IN   GEOMETRY. 

Then,  since  the  same  angle  Blue  has  been  taken 
from  equal  pairs  of  adjacent  angles,  what  do  you 
think  of  the  two  remainders,  Yellow,  Red  ? 

Suppose  the  lines  a  b  and  m  n  were  so  drawn  that 
the  angles  Yellow,  Red,  were  larger  or  smaller, 
would  they  still  be  equal  to  each  other  ? 

Then,  — 

All  vertical  angles  are  equal  to  each  other. 


THEOREMS   ILLUSTRATED. 


99 


Diagram    34. 


DEMONSTRATION. 

We  wish  to  prove  that 

All  vertical  angles  are  equal  to  each  other. 

Let  the  straight  lines  a  I,  m  n,  intersect  each  other 

at  the  point  c,  then  will  any  two  vertical  angles, 

as  Yellow,  Red,  be  equal  to  each  other. 
For  the  adjacent  angles  Yellow,  Blue,  are  equal  to 

two  right  angles.* 
The  adjacent   angles   Blue,  Red,  are  also  equal  to 

two  right  angles. 
Therefore  the   adjacent  angles  Yellow,  Blue,  are 

equal  to  the  adjacent  angles  Blue,  Red. 
If,  from  the  adjacent  angles  Yellow,  Blue,  we  take 

away  the   angle  Blue,  we   shall  have  left  the 

angle  Yellow. 

*  When  this  comparison  is  made,  let  the  pupil  look  at  the  right  angles  P  and  S. 


100  FIRST   LESSONS   IN    GEOMETRY. 

If,  from  the  adjacent  angles  Blue,  Red,  we  take 
away  the  same  angle  Blue,  we  shall  have  left  the 
angle  Red. 

Therefore  the  vertical  angles  Yellow,  Red,  are 
equal  to  each  other. 

TEST   QUESTIONS. 

When  you  say  that  the  adjacent  angles  Yellow, 
Blue,  are  equal  to  two  right  angles,  do  you 
know  it  because  you  see  it,  or  because  you  have 
proved  it  ? 

How  do  you  know  that  the  adjacent  angles  Blue, 
Red,  are  equal  to  two  right  angles  ? 

When  you  say  the  adjacent  angles  Yellow,  Blue, 
are  equal  to  the  adjacent  angles  Blue,  Red,  what 
axiom  do  you  use  ? 

What  same  thing  do  you  take  away  from  equals  ? 

From  what  equals  do  you  take  it  away  ? 

When  you  take  the  angle  Blue  from  the  adjacent 
angles  Yellow,  Blue,  what  is  the  remainder? 

When  you  take  the  same  angle  Blue  from  the 
adjacent  angles  Blue,  Red,  what  is  the  re- 
mainder ? 

What  do  you  find  true  of  the  two  remainders ? 

What  axiom  do  you  use  ? 


THEOEEMS   ILLUSTRATED. 


101 


Diagram   35. 
OTHER  METHODS   OF  DEMONSTRATION. 

The  adjacent  angles  Yellow,  Green,  are   equal  to 

what  ? 
The    adjacent    angles  Green,    Red,   are    equal    to 

what  ? 
Then  what    do    you    know  of   the  two    pairs   of 

adjacent  angles  Yellow,  Green,  and  Green,  Red  ? 
From   the    adjacent     angles  Yellow,    Green,   take 

away  the  angle  Green.     What  remains  ? 
From    the    adjacent    angles  Green,  Red,  take  the 

same  angle  Green.     What  remains  ? 
What  do  you  know  of  the  two  remainders? 
Why? 

What  axiom  do  you  use  ? 
In  the  last  lesson,  when  you  proved  the  vertical 

angles  Yellow,  Red,  equal  to    each    other,  you 

made    use  of  the  angle  Blue ;   now  prove  the 

same  two  angles  equal  by  means  of  the  angle 

Green. 


102  FIEST   LESSONS   IN   GEOMETRY. 

The  adjacent  angles  Blue,  Red,  are  equal  to  what  ? 
The  adjacent  angles  Red,  Green,  are  equal  to  what  ? 
Then  what  do  you  know  of  the  two  pairs  of 

adjacent  angles  Blue,  Red,  and  Red,  Green  ? 
From  the  adjacent  angles  Blue,  Red,  take  away 

the  angle  Red.      What  remains? 
From  the  adjacent  angles  Red,  Green,  take  away 

the  same  angle  Red.     What  remains? 
Then  what  do  3^011  know  of  the  two  remainders, 

Blue,  Green  ? 
Now  apply  the    preceding    demonstration  to  the 

vertical  angles  Blue,  Green. 
Prove  the  vertical    angles  Blue,  Green,  equal  to 

each  other  by  means  of  the  angle  Yellow. 


THEOREMS   ILLUSTRATED. 


103 


Diagram   36. 


TEST  LESSON. 
By  means  of  Fig.  A,  — 

1.  Prove  that  the  vertical  angles  Yellow,  Red,  are 
equal  to  each  other,  using  the  angle  Greert. 

2.  Prove  the  same  thing,  using  the  angle  Blue. 

3.  Prove  that  the  vertical  angles  Blue,  Green,  are 
equal  to  each  other,  using  the  angle  Yellow. 

4.  Prove  the  same  thirjg,  using  the  angle  Red. 


104  FIKST  LESSONS  IN   GEOMETRY. 

By  means  of  Fig.  B,  — 

5.  Prove  the  vertical  angles  Yellow,  Red,  equal  to 
each  other,  using  the  angle  Green. 

6.  Prove  the  same  thing,  using  the  angle  Blue. 

7.  Prove  the  vertical  angles  Green,  Blue,  equal  by 
means  of  the  angle  Red. 

8.  Prove  the  same  thing  by  means  of  the  angle 
Yellow. 

Go  through  the  preceding  eight  demonstrations 
again,  calling  the  angles  by  their  letters  instead 
of  by  their  colors. 

By  means  of  Fig.  C,  prove  that 

9.  a  c  n  equals  m  c  b,  by  means  of  a  c  m. 

10.  a  c  n  equals  m  c  b,  by  means  of  b  c  n. 

11.  a  c  m  equals  n  c  b,  by  means  of  a  c  n. 

12.  a  c  m  equals  n  c  #,  by  means  of  m  c  b. 
By  means  of  Fig.  D,  prove  that 

13.  m  c  a  equals  b  c  n,  by  means  of  a  c  n. 

14.  m  c  a  equals  b  c  n,  by  means  of  m  c  b. 

15.  m  c  b  equals  a  c  n,  by  means  of  m  c  a. 

16.  m  c  b  equals  a  c  n,  by  means  of  b  c  n. 


THEOREMS   ILLUSTRATED.  105 


Diagram    37. 


PROPOSITION!    III.      THEOREM. 

DEVELOPMENT  LESSON. 

In  the  above  diagram,  the  lines  a  b,  c  d,  are  paral- 
lel, and  are  intersected  by  the  line  ef  at  the 
points  m  and  n. 

The  angle  Red  measures  the  difference  of  direc- 
tion between  the  line  m  b  and  what  other  line  ? 

The  angle  Yellow  measures  the  difference  of  direc- 
tion between  the  line  n  d  and  what  other  line  ? 

Then,  as  the  lines  m  b  and  n  d  are  parallel,  must 
there  not  be  the  same  difference  of  direction 
between  them  and  the  line  ef? 

Then  can  there  be  any  difference  between  the 
angles  which  measure  those  equal  directions  ? 

Then  what  do  you  think  of  the  opposite  exterior 
and  interior  angles  Red,  Yellow  ? 


106  FIKST  LESSONS  IN  GEOMETKY. 

DEMONSTRATION. 

We  wish  to  prove  that 

Opposite  exterior  and  interior  angles  are  equal  to  each 
other. 

Let  the  straight  line  £/ intersect  the  two  parallel 

straight  lines  a  b\  c  d,  at  the  points  m  and  n. 
Then  will  any  two   opposite  exterior  and  interior 

angles,  as  Red,  Yellow,  be  equal  to  each  other. 
For  the    angle    Red    measures    the    difference    of 

direction  of  the  lines  m  b  and  ef. 
And  the  angle  Yellow  measures  the  difference  of 

direction  of  the  lines  n  d  and  ef. 
But  because  the  lines  m  b,  n  d,  are  parallel,  these 

differences  are  equal. 
Therefore    the    angles  which    measure    them    are 

equal ;  that  is, 
The  opposite    exterior  and   interior  angles   Red, 

Yellow,  are  equal  to  each  other. 


THEOREMS   ILLUSTRATED. 


107 


TEST  LESSON 

By  means  of  Fig.  A,  — 

1.  Prove   that  the  opposite  exterior  and  interior 
angles  Green,  Blue,  are  equal  to  each  other. 

2.  Prove  that  the  opposite   exterior  and  interior 
angles  Red,  Yellow,  are  equal  to  each  other. 

3.  Prove  the  opposite  exterior  and  interior  angles 
c  n  e,  a  m  n,  equal. 

4.  Prove  the  opposite  exterior  and  interior  angles 
e  n  d,  n  m  b,  equal. 


108  FIRST  LESSONS   IN   GEOMETRY. 

By  means  of  Fig.  B,  — 

5.  Prove  the  opposite  exterior  and  interior  angles 
e  m  a,  m  n  d,  equal. 

6.  Prove  the   opposite  exterior  and  interior  angles 
a  m  n,  d  nf^  equal. 

7.  Prove  the   opposite  exterior  and  interior  angles 
e  m  by  m  n  c,  equal. 

8.  Prove  the  opposite  exterior  and  interior  angles 
b  m  n,  c  nf,  equal. 


THEOREMS   ILLUSTRATED.  109 


—  I 


Diagram    39. 

PROPOSITION    IV.     THEOREM. 

DEVELOPMENT  LESSON. 

What  do  you  know  of  the  opposite  exterior  and 
interior  angles  Red,  Yellow? 

What  do  you  know  of  the  vertical  angles  Red, 
Green  ? 

Then  if  the  interior  alternate  angles  Green,  Yel- 
low, are  separately  equal  to  the  angle  Red,  what 
new  fact  do  you  know  ? 

What  axiom  do  you  employ  ? 

To  what  same  thing  did  you  find  two  things  equal  ? 

What  two  things  did  you  find  equal  to  it  ? 


110  FIRST   LESSONS   IN   GEOMET11Y. 


DEMONSTRATION. 

We  wish  to  prove  that 

Any  two  interior  alternate  angles  are  equal  to  each 
other. 

Let  the  straight  line  0/ intersect  the  two  parallel 

straight  lines  a  b,  c  d,  in  the  points  m  and  n. 
Then  will    any  two    interior   alternate  angles,  as 
*    Green,  Yellow,  be  equal  to  each  other. 
For  the  opposite  exterior  and  interior  angles  Red, 

Yellow,  are  equal. 

The  vortical  angles  Red,  Green,  are  also  equal 
Then  because  the  interior  alternate  angles  Green, 

Yellow,  are  separately  equal  to  the  angle   Red, 

they  are  equal  to  each  other. 


THEOREMS   ILLUSTRATED. 


Ill 


Diagram   4O. 


TEST  LESSON. 


What  do  you  know  of  the  vertical  angles  Green, 

Red,  in  Fig.  A  ? 
What  do  you  know  of  the  opposite  exterior  and 

interior  angles  Red,  Yellow  ? 
Then  if  the  interior  alternate  angles  Green,  Yel- 


112  FIRST  LESSONS   IN   GEOMETRY. 

low,  are  separately  equal  to  the  angle  Red,  what 
do  you  infer? 

By  means  of  Fig.  A, —  *- 

1.  Prove  that  the  interior  alternate  angles  Green, 
Yellow,  are  equal,  using  the  angle  Red. 

2.  Prove  the  same  angles  equal,  using  the  angle 
Blue. 

3.  Go    through    the    same    demonstrations   again, 
calling  the  angles  by  their  letters  instead  of  by 
their  colors. 

By  means  of  Fig.  B,  — 

4.  Prove  the   interior  alternate  angles  Red,  Blue, 
equal,  using  the  angle  Yellow. 

5.  Prove  the  same  angles  equal,  using  the  angle 
Green. 

6.  Go  through  the  same  two  demonstrations  again, 
naming  the  angles  by  their  letters  instead  of  by 
tfreir  colors. 

By  means  of  Fig.  C,  — 

7.  Prove  the  interior  alternate  angles  c  n  m,  n  m  b, 
equal,  using  the  angle  /  n  d. 

8.  Prove  the  same,  using  the  angle  a  m  e. 

9.  Prove  the  interior  alternate  angles  a  m  n,  m  n  d, 
equal,  using  the  angle  e  m  b.. 

10.  Prove  the  same,  using  the  angle  c  nf. 


THEOREMS   ILLUSTRATED.  113 


Diagram    41. 

PROPOSITION    V.     THEOREM. 

DEVELOPMENT  LESSON. 

What  do  you  know  of  the  opposite  exterior  and 

interior  angles  Red,  Yellow  ? 
What  do  you  know  of  the  vertical  angles  Yellow, 

Green  ? 
Then  if  the  exterior  alternate  angles  Red,  Green, 

are  separately  equal  to  the  angle  Yellow,  what 

new  thing  do  you  know  to  be  true  ? 
What  axiom  do  you  employ  ? 
To  what  same  thing  did  you  know  two  things  to  be 

equal  ? 

What  two  things  did  you  know  to  be  equal  to  it  ? 
Then  what  new  thing  did  you  find  to  be  true  ? 


114  FIItST  LESSONS   IN   GEOMETRY. 


DEMONSTRATION. 

We  wish  to  prove  that 

Any  two  exterior  alternate  angles  are  equal  to  each 
other. 

Let  the  straight  line  ef  intersect  the  two  parallel 

straight  lines  a  b,  c  d,  at  the  points  m  and  n. 
Then  will  any  two   exterior  alternate   angles,  as 

Red,  Green,  be  equal. 
For  the  opposite  exterior  and  interior  angles  Red, 

Yellow,  are  equal  to  each  other. 
And   the  vertical  angles  Yellow,  Green,  are  also 

equal  to  each  other. 
Then  because  the  exterior  alternate  angles  Red, 

Green,  are  separately  equal  to  the  angle  Yellow, 

they  are  equal  to  each  other. 


THEOREMS   ILLUSTRATED. 


115 


Diagram    42. 


TEST  LESSON. 


WJiat  do  you  know  of  the  opposite  exterior  and 

interior  angles  Yellow,  Red  ? 
What  do  you   know  of  the  vertical  angles  Red, 

Blue  ? 


11G  FIKST  LESSONS   IN   GEOMETRY. 

Then  if  the  exterior  alternate  angles  Yellow,  Blue, 
are  separately  equal  to  the  angle  Red,  what  do 
you  know  of  them  ? 

By  means  of  Fig.  A, — 

1.  Prove  that  the  exterior  alternate  angles  Yellow, 
Blue,  are  equal,  using  the  angle  Red. 

2.  Prove  the  same  thing,  using  the  angle  Green. 

3.  Go  through  the  same  demonstrations,  calling  the 
angles  by  their  letters. 

4.  Prove  the  exterior  alternate  angles  e  m  b,  c  nf, 
equal,  using  the  angle  a  m  n. 

5.  Prove  the  same,  using  the  angle  m  n  d. 

By  means  of  Fig.  B,  — 

6.  Prove  that  the  exterior  alternate  angles  c  m  e, 
f  n  b,  are  equal,  using  the  angle  n  m  d. 

7.  Prove  the  same,  using  the  angle  a  n  m. 

8.  Prove  the  exterior  alternate  angles  e  m  d,  a  nf, 
equal,  using  the  angle  c  m  n. 

9.  Prove  the  same,  using  the  angle  m  n  b. 


THEOREMS   ILLUSTHATKD. 


117 


Diagram    43. 

PROPOSITION    VI.     THEOREM. 

DEVELOPMENT  LESSON. 

What  do  you  know  of  the  interior  alternate  angles 
Yellow,  Red  ? 

If  to  the  angle  Green  you  add  the  angle  Yellow, 
what  is  the  sum  ? 

If  to  the  same  angle  G^een  you  add  the  equal 
angle  Red,  what  is  the  sum  ? 

Then,  having  added  equals  to  the  same  thing, 
what  do  you  think  of  the  two  sums,  —  the  ad- 
jacent angles  Green,  Yellow,  and  the  interior 
opposite  angles  Green,  Red  ? 

What  do  you  know  of  the  adjacent  angles  Green, 
Yellow,  and  the  right  angles  P,  S  ? 


118  FIRST  LESSONS   IN   GEOMETRY. 

Then  if  the  interior  opposite  angles  Green,  Red, 
and  the  two  right  angles  P,  S,  are  separately 
equal  to  the  adjacent  angles  Green,  Yellow, 
what  new  thing  do  you  know  ? 


DEMONSTRATION. 

We  wish  to  prove  that 

Any  two  interior  opposite  angles    are    equal    to   tivo 
right  angles. 

Let  the  straight  line  £/ intersect  the  two  parallel 

straight  lines  a  b,  c  d,  in  the  points  m  and  n. 
Then  will   any  two    interior   opposite    angles   be 

equal  to  two  right  angles! 
For  the  interior  alternate  angles  Yellow,  Red,  are 

equal. 
If  to  the   angle  Green  we  add  the  angle  Yellow, 

we  shall  have  the  adjacent  angles  Green,  Yellow. 
If  to  the  same  angle  Green  we  add  the  equal  angle 

Red,  we  shall  have  the  interior  opposite  angles 

Green,  Red. 
Then  the  adjacent  angles  Green,  Yellow,  are  equal 

to  the  interior  opposite  angles  Green,  Red. 
*But  the  adjacent  angles  Green,  Yellow,  are  equal 

to  two  right  angles. 
Then  because   the  interior  opposite  angles  Green, 

Red,  and  two  right  angles,  are  separately  equal 

to  the  two  adjacent  angles  Green,  Yellow,  they 

are  equal  to  each  other. 


THEOREMS   ILLUSTRATED.  119 

e 


Diaqram    44. 


TEST  LESSON. 
By  means  of  Fig.  A,  — 

1.  Prove  the  interior  opposite  angles  Green,  Yel- 
low, equal  to  two  right  angles,  using  the  angle 
Red. 

2.  Prove  the  same,  using  the  angle  Blue. 

3.  Prove  the  same,  using  the  angle  e  g  b. 

4.  Prove  the  same,  using  the 

(U'SIVBRSITT] 


120  FIRST   LESSONS   IN   GEOMETRY. 

5.  Go  through  the    same    demonstrations   again, 
naming  the  angles  by  their  letters  instead  of  by 
their  colors. 

6.  Prove  the  interior  opposite  angles  Red,  Blue, 
equal  to  two  right  angles,  using  the  angle  Yellow. 

7.  Prove  the  same,  using  the  angle  Green.  , 

8.  Prove  the  same,  using  the  angle  e  g  a. 

9.  Prove  the  same,  using  the  angle  c  hf. 

10.  Go  through  the  same   demonstrations    again, 
calling  the  angles  by  their  letters  instead  of  by 
their  colors. 

By  means  of  Fig.  B,  — 

11.  Prove  the  interior  opposite  angles  a  g  h,  g  h  c, 
equal  to  two  right  angles,  using  the  angle  g  h  d. 

12.  Prove  the  same,  using  the  angle  c  hf. 

13.  Prove  the  same,  using  the  angle  age. 

14.  Prove  the  interior  opposite  angles  b  y  h,  y  h  d, 
equal  to  two  right  angles,  using  the  angle  a  g  h. 

15.  Prove  the  same,-  using  the  angle  e  g  b. 

16.  Prove  the  same,  using  the  angle  f  h  d.   . 
Compare   the  angles  Yellow,  Green,  each  with  its 

exterior  opposite  angle,  and  see  if  you  can  prove 
that  the  exterior  opposite  angles  e  g  b,  f  h  d,  are 
also  equal  to  two  right  angles. 


THEOREMS   ILLUSTRATED.  121 


Diagram    45. 


PROPOSITION   VII.     THEOREM. 

DEVELOPMENT  LESSON. 

Suppose  we  do  not  know  whether  the  lines  a  #,  c  d, 

are  parallel,  or  not ; 
But,  by  measuring,  we  find  that  the  interior  angles 

Blue,  Yellow,  on  the  same  side  of  the  secant* 

line  ef,  are  equal  to  two  right  angles: 
The  adjacent  angles  Blue,  Red,  are  equal  to  what? 
Then,  if  the  interior  angles  Blue,  Yellow,  are  equal 

to  two  right  angles, 
And  the  adjacent  angles  Blue,  Red,  are  also  equal 

to  two  right  angles, 
What  do  you  infer? 
From  the  interior  angles  Blue,  Yellow,  take  away 

the  angle  Blue  :  what  remains  ? 
From  the  adjacent  angles  Blue,  Red,  take  away  the 

same  angle  Blue  :  what  remains  ? 

*  "  Secant"  means  "  cutting." 


122  FIRST  LESSONS  IN   GEOMETRY. 

What  do  you  know  of  the  two  remainders  ? 

The  angle  Red  measures  the  direction  of  the  line 

g  b  from  what  line  ? 
The  equal  angle  Yellow  measures  the  direction  of 

the  line  h  d  from  what  line  ? 

Then  if  the  lines  g  b,  h  d,  have  the  same  direction 
.  from  the  line  ef,  what  do  you  call  them? 


Diagram    46. 

DEMONSTRATION. 

We  wish  to  prove,  that, 

If  a  straight  line  intersects  two  other  straight  lines  so 
that  two  interior  angles  on  the  same  side  of  the 
intersecting  line  are  equal  to  two  right  angles9  the 
two  lines  are  parallel. 

Let  the  straight  line  0/ intersect  the  two  straight 
lines  a  b,  c  d,  in  the  points  g  and  h,  so  that  the 
angles  Red,  Blue,  are  equal  to  two  right  angles. 


THEOREMS   ILLUSTRATED.  123 

Then  will  the  lines  a  b,  c  d,  be  parallel. 

For  the  angles  Red,  Blue,  are  supposed  equal  to 

two  right  angles. 
The  adjacent  angles  Red,  Green,  are  known  to  be 

also  equal  to  two  right  angles. 
Then  the  interior  angles  Red,  Blue,  are  equal  to 

the  adjacent  angles  Red,  Green. 
If  from   the  interior  angles  Red,  Blue,  we  take 

away  the  angle   Red,   we   have  left  the   angle 

Blue. 
If  from  the  adjacent   angles  Red,  Green,  we  take 

the  same  angle  Red,  we  shall  have  left  the  angle 

Green. 

Then  the  angle  Blue  is  equal  to  the  angle  Green. 
But  the  angle  Blue  measures  the  direction  of  the 

line  h  d  from  the  line  ef. 
And  the  angle  Green  measures  the  direction  of  the 

line  g  b  from  the  line  ef. 
Then  the  lines  g  b,  h  d,  have  the  same  direction, 

and  are  parallel. 

TEST  LESSON. 

1.  Prove  the  same  without  the  colors. 

2.  Prove  the  same,  using  the  angle  /  h  d. 

3.  Prove  the  same,  supposing    the    angles   a  g  h, 
g  h  c,  equal  to  two  right  angles,  and  using  the 
angle  age. 

4.  Prove  the  same,  using  the  angle  c  hf. 

See  Note  E,  Appendix. 


124  FIllST   LESSONS   IN   GEOMETRY. 


PROPOSITION    VIII.     THEOREM. 

The  following  demonstration  is  very  easy.  Read 
it  once,  and  see  if  you  can  go  through  it  with- 
out a  second  reading  :  — 

DEMONSTRATION. 

We  wish  to  prove  that 

The  sum  of  any  two  sides  of  a  triangle 
is  greater  than  the  third  side. 

Let  the  figure  a  b  c  be  a  triangle, 
then  will  the  sum  of  any  two  sides, 
as  a  c,  c  b,  be  greater  than  the  third 
side  a  b. 

For  the  straight  line  a  b  is  the  short- 
est    distance    between     the    two  c 
points  a   and  £,    and  is    therefore 
less  than  the  broken  line  a  c  b. 


PROPOSITION    IX.     PROBLEM. 

The   following   solution  is  so  easy  that  you  will 
understand  it  at  once  :  — 

We  wish 

To  construct  an  equilateral  triangle  on  a  given  straight 
line. 


PROBLEM  ILLUSTRATED. 


125 


SOLUTION. 

Let  a  b  be  the  given  line. 

With  the  point  a  as  a  centre,  and  a  b  as  a  radius, 

draw  the  circumference  of  the  circle,  or  a  part 

of  one. 
With  the  point  b  as  a  centre,  and  the  same  radius 

a  by  draw  another  circumference,  or  a  part  of 

one. 
From  the  point  c9  in  which  the  circumferences  or 

arcs  intersect,  draw  the  straight  lines  a  c  and  b  c. 
Now,  because  the  lines  a  b  and  a  c  are  radii  of  the 

same  circle,  they  are  equal. 
And,  because  the  lines  a  b  and  b  c  are  radii  of  the 

same  circle,  they  are  also  equal. 
Then,  because  the  two  lines  a  c,  b  c,  are  separately 

equal  to  the  line  a  b,  they  are  equal  to  each 

other,  and  the  triangle  is  equilateral. 


126  PIBST  LESSONS  IN  GEOMETKY. 


PROPOSITION    X.     THEOREM. 

DEVELOPMENT    LESSON. 

• 

Let  the  figure  a  b  c  be  a  triangle. 

Produce  the  side  a  c  to  d. 

We  have  now  another  angle,  b  c  d,  and  we  wish  to 

find  out  if  it  is  equal  to  any  of  the  angles  of 

the  triangle. 

From  the  point  c  draw  the  line  c  e  parallel  to  a  b. 
Because   the  straight  line  a  ^intersects  the  two 

parallels  a  b,  c  e>  the   angle  a  is   equal  to  what 

other  angle  ? 
Because  the   straight  line  b  c  intersects  the  two 

parallels  a  b,  c  e,  the  angle  b  is  equal   to  what 

other  angle  ? 
Then   the  angles  a  and  b  are  equal  to  what  two 

angles  ? 
How  does  the  angle  bed  compare  with  the  angles 

b  v'e,  e  c  d? 
Then,  if  the  angles  a  and  5,  on  the  one  hand,  and 

the  angle  b  c  d,  on  the  otheryare  separately  equal 

to  the  angles  b  c  e,  e  c  d, 
What  have  you  found  out? 


THEOREMS   ILLUSTRATED.  127 

What  axiom  have  you  just  employed  ? 

To  what  same  thing  have  you  found  two   other 

things  equal  ? 
What  two  things  did  you  find  equal  to  it? 


DEMONSTRATION. 
We  wish  to  prove,  that, 

If  any  side  of  a  triangle  be  produced,  the  new  angle 
formed  ivlll  be  equal  to  the  sum  of  the  angles  that 
are  not  adjacent  to  it. 

Let  a  b  c  be  a  triangle. 

Produce  the  side  a  c  to  d ;  then  will  the  new  angle 

b  c  d  be  equal  to  the  sum  of  the  angles  a  and  b. 
:Eor  from  the  point  c  draw  c  e  parallel  to  a  b. 
Then,  because  the  straight  line  a  d  intersects  the 

two  parallels  a  b,  c  e,  in  the  points  a  and  .c9. 
The   opposite  exterior  and  interior  angles  a  and 

e  c  d  are  equal  to  each  other. 
And   because   the   straight  line  be  intersects  the 

same  parallels  in  the  points  b  and  c," - 
The  interior  alternate  angles  b  and  bee  are  equal. 
Then  the  angles  a  and  b  of  the  triangle  are  equal 

to  the  angles  bee  and  e  c  d. 
But  the   new'  angle  be  d  is  equal  to  the  angles 

b  c  e,  e  c  d. 
Then  because  the  new  angle  b  c  d,  and  the  angles 

a  and  b  are  separately  =  to  the,  angles  b'c:e, 

e  c  d,  they  are  equal  to  each  other. 


128  FIRST  LESSONS  IN  GEOMETRY. 


PROPOSITION    XL     THEOREM. 

DEVELOPMENT  LESSON. 

Let  the  figure  a  b  c  be  a  triangle. 

Produce  the  side  a  c  to  d. 

By  the  last  theorem,  the  angle  I  c  d  is  equal  to 

what  .angles  of  the  triangle  ? 
What  angle  must  we  add  to  these  angles  to  make 

up  the  three  angles  of  the  triangle  ? 
If  we  add  the  same  angle  to  the  angle  I  c  d,  what 

adjacent  angles  do  we  get  ? 
Then  the  three  angles  of  the  triangle,  #,  b,  and  c9 

are  equal  to  what  two  angles  ? 
But  the  adjacent  angles  a  c  b  and  bed  are  equal  to 

what? 
Then,  because   the  three  angles  of  the  triangle, 

a,  J,  and  e,  and  two  right  angles,  are  separately 

equal  to  the  two  adjacent  angles  c  and  b  c  d, 
What  new  thing  have  you  found  out  ? 


THEOREMS   ILLUSTRATED.  129 


DEMONSTRATION. 

We  wish  to  prove  that 

The  three  angles  of  any  triangle   are   equal   to  two 
right  angles* 

Let  the  figure  ale  be  a  triangle ;  then  will  the 

sum  of  the  angles  a,  #?  and  c,  be  equal  to  two 

right  angles. 

For,  produce  the  side  a  c  to  d, 
The  new  angle  b  c  d  is  equal  to   the  sum  of  the 

angles  a  and  b. 
If  to  the  angles  a  and  b  we  add  the  angle  c,  we 

shall  have  the  three  angles  of  the  triangle. 
If  to  the  angle  b  c  d  we  add  the  same  angle  c,  we 

shall  have  the  adjacent  angles  c  and  bed. 
Then  the  three  angles  of  the  triangle  a,  b,  c,  are 

equal  to  the  adjacent  angles  c  and  b  c  d. 
But  the   adjacent  angles  c  and  b  c  d  are  equal  to 

two  right  angles. 
Then,  because  the  three  angles  of  the  triangle  are 

equal  to   the  adjacent  angles  c  and  b  c  d,  they 

are  equal  to  two  right  angles. 


130  FIRST  LESSONS   IN   GEOMETRY. 


PROPOSITION    XII.     THEOREM. 

DEVELOPMENT  LESSON. 

Let  the  Fig.  A  B  C  D  be  a  parallelogram. 

Produce  the  side  C  D  to  F. 

Because  the  straight  line  B  D  intersects  the  paral- 
lels A  B  and  C  F,  the  angle  B  is  equal  to  what 
other  angle  ? 

Because  the  straight  line  C  F  intersects  the  paral- 
lels A  C  and  B  D,  the  angle  C  is  equal  to  what 
other  angle  ? 

Then  what  follows  from  this  ? 

To  what  angle  did  you  find  two  others  equal  ? 

What  two  angles  did  you  find  equal  to  it  ? 

What  axiom  do  you  think  of? 

See  if  you  can  go  through  the  demonstration  with 
out  reading  it  even  once. 


DEMONSTRATION. 
We  wish  to  prove  that 

The  opposite  angles  of  a  parallelogram  are  equal  to 
each  other. 

Let  the  Fig.  A  B  C  D  be  a  parallelogram. 


THEOREMS  ILLUSTRATED.  131 

Then  will  any  two  opposite  angles,  as  B  and  C,  be 
equal  to  each  other. 

For  produce  the  line  C  D  to  F. 

Because  the  straight  line  B  D  meets  the  two  par- 
allels A  B  and  C  F, 

The  interior  alternate  angles  B  and  E  are  equal  to 
each  other. 

Because  the  straight  line  C  F  meets  the  two  par- 
allels B  D  and  A  C, 

The  opposite  exterior  and  interior  angles  C  and  E 
are  equal  to  each  other. 

Then,  because  the  angles  B  and  C  are  separately 
equal  to  the  angle  E,  they  are  equal  to  each 
other. 


1.  Prove   the  same  by  producing  the  line  A  B 
towards  the  left. 

2.  Prove  the    same   by  producing  the  line  B  D 
downwards. 

3.  Prove  the  angles  A  and  D  equal  to  each  other 
by  producing  the  line  C  D  towards  the  left. 

4.  Prove  the   same  by  producing  the  line  D   B 
upwards. 

5.  See  if  you  can  prove  the  same  by  drawing  a 
diagonal  through  the  points  A  and  D. 


132  FIRST  LESSONS   IN   GEOMETRY. 


\f 


PROPOSITION   XIII.     THEOREM. 

DEVELOPMENT  LESSON. 

In  these  two  triangles  we  have  tried  to  make  the 
side  a  b  of  the  one  equal  to  the  side  d  e  of  the 
other ;  the  side  a  c  of  the  one  equal  to  the  side 
d  /of  the  other;  and  the  included  angle  b  a  c 
of  the  one  equal  to  the  included  angle  e  df  of 
the  other. 

We  now  wish  to  find  out  if  the  third  side  b  c  of 
the  one  is  equal  to  the  third  side  e  f  of  the 
other,  and  if  the  two  remaining  angles  b  and  c 
of  the  one  are  equal  to  the  two  remaining  angles 
e  and  /  of  the  other. 

Suppose  we  were  to  cut  the  triangle  def  out  of 
the  page,  and  place  it  upon  the  triangle  a  b  c,  so 
that  the  line  d  e  should  fall  upon  the  line  a  b, 
and  the  point  d  upon  the  point  a. 

As  the  line  d  e  is  equal  to  the  line  a  b,  upon  what 
point  will  the  point  e  fall? 

If  the   angle  e  df  were  less   than  the  angle  b  a  c^ 


THEOREMS   ILLUSTRATED.  133 

would  the  line  d  f  fall  within  or  without  the 
triangle  ? 

If  the  angle  e  d  f  were  greater  than  the  angle 
b  a  c,  where  would  the  line  df  fall  ? 

Since  the  angle  a  is  equal  to  d,  w&ere,  then,  must 
the  line  df  fall  ? 

As  the  line  d  f  is  equal  to  the  line  a  c,  upon  what 
point  will  the  point  /  fall  ? 

Then,  if  the  point  e  falls  upon  the  point  b,  and  the 
point  /  upon  the  point  c,  where  will  the  line  ef 
fall?  " 

Now,  because  the  three  sides  of  the  triangle  d  ef 
exactly  fall  upon  the  three  sides  of  the  triangle 
a  b  c,  we  say  the  tivo  magnitudes  coincide  throughout 
their  whole  extent,  and  are  therefore  equal. 

What  three  parts  of  the  triangle  a  b  c  did  we  sup- 
pose to  be  equal  to  three  corresponding  parts  of 
the  triangle  d  ef  before  we  placed  one  upon  the 
other. 

What  line  of  the  one  do  vtQ  find  equal  to  a  line  in 
the  other  ? 

What  two  angles  of  the  one  do  we  find  equal  to 
two  angles  in  the  other  ? 

What  do  you  think  of  the  areas  of  the  triangles  ? 


134  FIRST  LESSONS  IN  GEOMETKY. 

d 


U  e  f 

DEMONSTRATION. 
We  wish  to  prove,  that, 

If  two  triangles  have  two  sides ,  and  the  included  angle 
of  the  one  equal  to  two  sides  and  the  included 
angle  of  the  other,  each  to  each9  the  two  triangles 
are  equal  in  all  respects. 

Let  the  triangles  a  b  c  and  d  ef  have  the  side  a  b 
of  the  one  equal  to  the  side  d  e  of  the  other; 
the  side  a  c  of  the  one  equal  to  the  side  df  of 
the  other- ;  and  the  included  angle  b  a  c  of  the 
one  equal  to  the  included  angle  e  d  f  of  the 
other,  each  to  each  ;  then  will  the  two  triangles 
be  equal  in  all  their  parts. 

For,  place  the  triangle  d  e  f  upon  the  triangle  a  b  c, 
so  that  the  line  d  e  shall  fall  upon  the  line  a  b, 
with  the  point  d  upon  the  point  a. 

Because  the  line  d  e  is  equal  to  the  line  a  b,  the 
point  e  will  fall -upon  the  point  b. 

Because  the  angle  e  df  is  equal  to  the  angle  b  a  c, 
the  line  df  will  fall  upon  the  line  a  c. 

Because  the  line  df  is  equal  to  the  line  a  c,  the 
point/  will  fall  upon  the  point  c. 


THEOREMS  ILLUSTRATED.  135 

Then,  because  the  point  e  is  on  the  point  b,  and  the 
point  f  on  the  ppint  c9  the  line  ef  will  coincide 
with  the  line  b  c,  and  the  two  triangles  will  be 
found  equal  in  all  their  parts ; 

That  is,  the  angle  e  is  found  to  be  equal  to  the 
angle  #,  the  angle  /  to  the  angle  c,  the  line  ef 
to  the  line  b  c,  and  the  area  of  the  triangle 
a  b  c  to  the  area  of  the  triangle  d  ef. 


136  FIRST  LESSONS   IN    GEOMETRY. 

d 


PROPOSITION    XIV.     THEOREM. 

DEVELOPMENT  LESSON. 

In  these  two  triangles  we  have  tried  to  make  the 
angle  b  of  the  one  equal  to  the  angle  e  of  the 
other ;  the  angle  c  of  the  one  equal  to  the  angle 
/  of  the  other;  and  the  included  side  b  c  of 
the  one  equal  to  the  included  side  ef  of  the 
other. 

We  now  wish  to  find  out  if  the  remaining  angle 
a  of  the  one  is  equal  to  the  remaining  angle  d 
of  the  other,  and  if  the  two  remaining  sides  a  b 
and  a  c  of  the  one  are  equal  to  the  two  remain- 
ing sides  d  e  and  df  of  the  other. 

Suppose  we  were  to  cut  the  triangle  d  ef  out  of 
the  page  and  place  it  upon  the  triangle  a  b  c,  so 
that  the  line  ef  shall  fall  upon  the  line  b  c,  with 
the  point  e  upon  the  point  b. 

Because  the  line  efis  equal  to  the  line  b  c,  upon 
what  point  will  the  point /fall? 

Because  the  angle  e  is  equal  to  the  angle  b,  where 
will  the  line  *rf  fall? 


THEOREMS  ILLUSTRATED.  137 

Because  the  angle /is  equal  to  the  angle  c,  where 
will  the  line  df  fall  ? 

Then,  if  the  line  d  e  falls  upon  the  line  a  l>,  and 
the  line  d  f  upon  the  line  a  c,  where  will  the 
point  d  fall  ? 

Now  because  the  three  sides  of  the  triangle  d  ef 
exactly  fall  upon  the  three  sides  of  the  triangle 
a  b  c,  we  say  the  tivo  magnitudes  coincide  throughout 
their  ivhole  extent,  and  are  therefore  equal 

Suppose  the  angle  e  were  greater  than  the  angle  b, 
would  the  line  e  d  fall  within  or  without  the  tri- 
angle ? 

If  it  were  less,  where  would  the  line  fall  ? 

Why  does  the  line  d  e  fall  exactly  upon  the  line  a  b  ? 


138  FIKST  LESSONS   IN   GEOMETRY. 


b  c  e  f 

DEMONSTRATION. 

We  wish  to  prove  that, 

If  two  triangles  have  two  angles,  and  the  included  side 
of  the  one  equal  to  two  angles  and  the  included 
side  of  the  other,  each  to  each9  the  two  triangles 
are  equal  to  each  oilier  in  all  respects. 

Let  the  triangles  a  b  c  and  d  ef  have  the  angle  b  of 
the  one  equal  to  the  angle  e  of  the  other;  the 
angle  c  of  the  one  equal  to  the  angle /of  the 
other ;  and  the  included  side  b  c  of  the  one  equal 
to  the  included  side  e  f  of  the  other,  each  to 
each ;  then  will  the  two  triangles  be  equal  in  all 
their  parts. 

For  place  the  triangle  d  ef  upon  the  triangle  a  b  C, 
so  that  the  line  e  f  shall  fall  upon  the  line  b  c, 
with  the  point  e  upon  the  point  b. 

Because  the  line  e  f  is  equal  to  the  line  b  c,  the 
point/ will  fall  upon  the  point  c. 

Because  the  angle  e  is  equal  to  the  angle  b,  the  line 
e  d  will  fall  upon  the  line  b  a,  and  the  point  d 
will  be  somewhere  in  the  line  b  a. 


THEOREMS   ILLUSTRATED.  139 

Because  the  angle  £  is  equal  to  the  angle  c,  the 
Ymefd  will  fall  upon  the  line  c  a,  and  the  point 
d  will  be  somewhere  in  the  line  c  a. 

Then,  because  the  point  d  is  in  the  two  lines,  ~b  a 
and  c  a,  it  must  be  in  their  intersection,  or  upon 
the  point  a. 

And,  as  the  two  triangles  coincide  throughout  their 
whole  extent,  they  are  equal  in  all  their  parts. 

That  is,  the  angle  a  is  found  to  be  equal  to  the 
angle  d ;  the  side  b  a  to  the  side  £  d ;  the  side 
c  a  to  the  side/rf;  and  the  area  of  the  triangle 
a  1)  c  to  the  area  of  the  triangle  d  ef. 


140  FIRST   LESSONS   IN   GEOMETRY. 

b 


PROPOSITION    XV.     THEOREM. 

DEMONSTRATION. 

We  wish  to  prove  that 

The  opposite  sides  of  any  parallelogram  are  equal. 

Let  the  figure  a  b  c  d  be  a  parallelogram ;  then  will 
the  sides  a  b  and  c  d  be  equal  to  each  other ; 
likewise  the  sides  a  d  and  b  c. 

For,  draw  the  diagonal  b  d. 

Because  the  figure  is  a  parallelogram,  the  sides 
a  b  and  d  c  are  parallel,  and  the  interior  alter- 
nate angles  n  and  o  are  equal. 

Because  the  figure  is  a  parallelogram,  the  interior 
alternate  angles  r  and  m  are  equal. 

Then  the  two  triangles  a  d  b,  b  d  c,  have  two  angles 
and  the  included  side  of  the  one  equal  to  two 
angles  and  the  included  side  of  the  other,  each 
to  each,  and  are  therefore  equal ; 

And  the  side  a  b  opposite  the  angle  m  is  equal  to 
the  side  c  d  opposite  the  equal  angle  r  ; 

And  the  side  a  d  opposite  the  angle  n  is  equal  to 
the  side  b  c  opposite  the  equal  angle  o. 

TEST. 
Prove  the  same  by  drawing  a  diagonal  from  a  to  c. 


THEOREMS   ILLUSTRATED. 
C 


141 


D 


PROPOSITION    XVI.     THEOREM. 

DEVELOPMENT  LESSON. 

Suppose  A  B  to  be  a  straight  line,  and  c  any  point 
out  of  it. 

From  the  point  c  draw  a  perpendicular  c  F  to 
A  B. 

Let  us  see  if  this  perpendicular  is  not  shorter  than 
any  other  line  we  can  draw  from  the  same  point 
to  the  same  line. 

Draw  any  other  line  from  c  to  A  B  as  c  E. 

Now,  as  c  E  is  any  line  whatever  other  than  a  per- 
pendicular, if  we  find  that  the  perpendicular 
c  F  is  shorter  than  it  we  must  conclude  that  it  is 
the  shortest  line  that  can  be  drawn  from  c  to  A  B. 

Produce  c  F  until  F  D  is  equal  to  c  F,  and  then  join 
E  and  D. 


142  FIRST   LESSONS   IN    GEOMETRY. 

In  the  triangles  EEC,  E  F  D,  what  two  sides  were 
drawn  equal? 

What  line  is  a  side  to  each  ? 

How  great  an  angle  is  c  F  E  ? 

What  is  a  right  angle  ? 

Then  how  do  the  angles  c  F  E  and  E  F  D  compare 
with  each  other  ? 

If  the  two  triangles  E  F  c,  E  F  D,  have  the  side  c  F 
of  the  one  equal  to  the  side  F  D  of  the  other, 
the  side  E  F  common  to  both,  and  the  included 
angle  E  F  c  of  the  one  equal  to  the  included 
angle  E  F  D  of  the  other,  each  to  each,  what  do 
you  irtfer  ? 

Then  what  third  side  of  the  one  have  you  found 
equal  to  a  third  side  of  the  other  ? 

c  E  is  what  part  of  the  broken  line  c  E  D? 

c  F  is  what  part  of  the  line  CD? 

Which  is  shorter,  the  straight  line  c  D,  or  the 
broken  line  c  E  D  ? 

Then  how  does  the  half  of  c  D  or  c  F  compare  with 
the  half  of  c  E  D  or  c  E  ? 

If  c  E  is  any  line  whatever  other  than  a  perpen- 
dicular, what  may  we  now  say  of  tlie  perpen- 
dicular from  the  point  c  to  the  straight  line  A  13  ? 


THEOREMS  ILLUSTRATED. 
C 


143 


D 


DEMONSTRATION. 

We  wish  to  prove  that 

A  perpendicular  is  the  shortest  distance  from  a  point 
to  a  straight  line. 

Let  A  B  be  a  straight  line,  and  c  a  point  out  of  it; 
then  will  the  perpendicular  c  E  be  the  shortest 
line  that  can  be  drawn  from  the  point  to  the 
line. 

For  draw  any  other  line  from  c  to  A  B,  as  c  F. 

Produce  c  E  until  E  D  equals  c  E,  and  join  F  D.W 

The  two  triangles  F  E  c,  F  E  D,  have  the  side  c  E  of 
the  one  equal  to  the  side  E  D  of  the  other,  the 
side  F  E  common,  and  the  included  angle  F  E  .c  of 
the  one  equal  to  the  included  angle  F  E  D  of  the 
other,  they  are  therefore  equal,  arid  the  side  c  F 
eq'::*ls  the  side  F  D. 


144: 


FIRST   LESSONS   IN   GEOMETRY. 


But   the   straight  line   c  D  is  the  shortest  distance 

between    the    two    points    c  D  ;    therefore    it    is 

shorter  than  the  broken  line  c  F  D. 
Then  c  E,  the  half  of  c  D,  is  shorter  than  c  F,  the 

half  c  F  D. 
And,  as  c  F  is  any  line  other  than  a  perpendicular, 

the  perpendicular  c  E  is  the  shortest  line   that 

can  be  drawn  from  c  to  A  B. 


PROPOSITION    XVII.     THEOREM. 

DEMONSTRATION. 
We  Wish  to  prove  that 

A  tangent  to  a  circumference  *is  perpendicular  to  a 
radius  at  the  point  of  contact. 

Let  the  straight  line  A  B  be  tangent  at  the  point  D 
to  the  circumference  of  the  circle  whose  centre 
is  c. 


THEOREMS    ILLUSTRATED.  145 

Join  the  centre  c  with  the  point  of  contact  D5  the 
tangent  will  be  perpendicular  to  the  radius  c  D. 

For  draw  any  other  line  from  the  centre  to  the 
tangent,  as  c  F. 

As  the  point  D  is  the  only  one  in  which  the  tangent 
touches  the  circumference,  any  other  point,  as  F, 
must  be  without  the  circumference. 

Then  the  line  c  F,  reaching  -beyond  .the  circumfer- 
ence, must  be  longer  than  the  radius  c  D,  which 
would  reach  only  to  it ;  therefore  c  D  is  shorter 
than  any  other  line  which  can  be  drawn  from 
the  point  c  to  the  straight  line  A  B  ;  therefore  it 
is  perpendicular  to  it. 


140 


FIKST  LESSONS  IN  GEOMETRY. 


PROPOSITION    XVIII.      THEOREM. 

DEMONSTRATION. 

A 

We  wish  to  prove,  that, 

In  any  isosceles  triangle,  the 
angles  opposite  the  equal 
sides  are  equal. 

Let  the  triangle  A  B  c  be  isos- 
celes, having  the  side  A  B  Bt 


equal  to  the  side  A  c ;  then  will  the  angle  B, 
opposite  the  side  A  c,  be  equal  to  the  angle  c, 
opposite  the  equal  side  A  B. 

For  draw  the  line  A  D  so  as  to  divide  the  angle  A 
into  two  equal  parts,  and  let  it  be  long  enough 
to  divide  the  side  B  c  at  some  point  as  D. 

Now  the  two  triangles  A  D  B,  A  D  c,  have  the  side 
A  B  of  the  one  equal  to  the  side  A  c  of  the  other, 
the  side  A  D  common  to  both,  and  the  included 
angle  B  A  D  of  the  one  equal  to  the  included 
angle  c  A  D  of  the  other ;  therefore  the  two  tri- 
angles are  equal  in  all  respects,  and  the  angle 
B,  opposite  the  side  A  c,  is  equal  to  the  angle  c, 
opposite  the  .side  A  B. 


THEOREMS  ILLUSTRATED. 


147 


D         B 


PROPOSITION   XIX.      THEOREM. 

DEMONSTRATION. 

We  wish  to  prove  that, 

If  two  triangles  have  the  three  sides  of  the  one  equal 
to  the  three  sides  of  the  other ,  each  to  each,  they 
are  equal  in  all  their  parts. 

Let  the  two  triangles  ABC,  ADC,  have  the  side 
A  B  of  the  one  equal  to  the  side  A  D  of  the  other ; 
the  side  B  c  of  the  one  equal  to  the  side  D  c  of 
the  other,  and  the  third  side  likewise  equal; 
then  will  the  two  triangles  be  equal  in  all  their 
parts. 

For  place  the  two  triangles  together  by  their 
longest  side,  and  join  the  opposite  vertices 
B  and  D  by  a  straight  line. 

Because  the  side  A  B  is  equal  to  the  side  A  D,  the 
triangle  B  A  D  is  isosceles,  and  the  angles  A  B  D, 
A  D  B,  opposite  the  equal  sides  are  equal. 


148  FIKST  LESSONS   IK   GEOMETRY. 

Because  the  side  B  c  is  equal  to  the  side  D  c,  the 
triangle  B  c  D  is  isosceles,  and  the  angles  c  B  D, 
c  D  B,  opposite  the  equal  sides  are  equal. 

If  to  the  angle  A  B  D  we  add  the  angle  D  B  c,  we 
shall  have  the  angle  ABC. 

And  if  to  the  equal  of  A  B  D,  that  is,  A  D  B,  we  add 
the  equal  of  D  B  c,  that  is,  B  D  c,  we  shall  have 
the  angle  ADC. 

Therefore  the  angle  A  B  c  is  equal  to  the  angle 
ADC. 

Then  the  two  triangles  A  B  c,  A  D  c,  have  two  sides, 
and  the  included  angle  of  the  one  equal  to  two 
sides  and  the  included  angle  of  the  other,  each 
to  each,  and  are  equal  in  all  their  parts ;  that  is, 
the  three  angles  of  the  one  are  equal  to  the 
three  angles  of  the  other,  and  their  areas  are 
equal. 


THEOREMS  ILLUSTRATED. 


149 


PROPOSITION    XX.     THEOREM. 

DEMONSTRATION. 

We  wish  to  prove  that 

An  angle  at  the  circumference  is  measured  by  half 
the  arc  on  which  it  stands. 

Let  B  A  D  be  an  angle  whose  vertex  is  in  the  cir- 
cumference of  the  circle  whose  centre  is  c;  then 
will  it  be  measured  by  half  the  arc  B  D. 

For  through  the  centre  draw  the  diameter  A  E,and 
join  the  points  c  and  B. 

The  exterior  angle  E  c  B  is  equal  to  the  sum  of  the 
angles  B  and  BAG. 

Because  the  sides  c  A,  c  B,  are  radii  of  the  circle, 
they  are  equal,  the  triangle  is  isosceles,  the 
angles  B  and  BAG  opposite  the  equal  sides  are 
equal,  and  the  angle  B  A  c  is  half  of  both. 

Then,  because  the  angle  B  A  c  is  half  of  B  and  BAG, 
it  must  be  half  of  their  equal  E  c  B. 


150  FIRST  LESSONS   IN   GEOMETRY. 

But  E  c  B,  being  at  the  centre,  is  measured  by  B  E  ; 

then  half  of  it,  or  B  A  c,  must  be  measured  by 

half  B  E. 
In  like  manner,  it  may  be  proved  that  the  angle 

c  A  D  is  measured  by  half  E  D. 
Then,  because  BAG  is  measured  by  half  B  E,  and 

c  A  D  by  half  E  D,  the  whole  angle  BAD  must  be 

measured  by  half  the  whole  arc  B  D. 


SECOND   CASE. 

Suppose  the  angle  were  wholly  on  one  side  qf  the 

the  centre,  as  F  A  B. 

Draw  the  diameter  A  E  and  the  radius  B  c  as  before. 
Prove  that  the  angle  B  A  E  is  measured  by  half  the 

arc  B  E. 
Draw  another  radius  from  c  to  F,  and  prove  that 

F  A  E  is  measured  by  half  the  arc  F  E. 
Then,  because  the  angle  F  A  E  is  measured  by  half 

the  arc  F  E,  and  the  angle  B  A  E  is  measured  by 

half  the  arc  B  E, 
The    difference  of  the    angles,  or  FAB,  must   be 

measured  by  half  the  difference  of  the  arcs,  or 

half  of  F  B. 


THEOREMS   ILLUSTRATED. 


151 


PROPOSITION    XXI.     THEOREM. 

DEMONSTRATION. 

We  wish  to  prove  that 

Parallel  chords  intercept  equal  arcs  of  the  circumfer- 
ence. 

Let  the  chords  A  B,  c  D,  be  parallel ;  then  will  the 
intercepted  arcs  A  c  and  B  D  be  equal. 

For  draw  the  straight  line  B  c. 

Because  the  lines  A  B  and  c  D  are  parallel,  the  inte- 
rior alternate  angles  A  B  c,  B  c  D,  are  equal. 

But  the  angle  A  B  c  is  measured  by  half  the  arc 
A  cj 

And  the  angle  B  c  D  is  measured  by  half  the  arc 
B  D: 

Then,  because  the  angles  are  equal,  the  half  arcs 
which  measure  them  must .  be  equal,  and  the 
whole  arcs  themselves  must  be  equal. 


152 


FIKST  LESSONS   IN  GEOMETRY. 


7) 


PROPOSITION    XXII.     THEOREM. 

DEMONSTRATION. 
We  wish  to  prove  that 

The  angle  formed  l>y  a  tangent  and  a  chord  meeting 
at  the  point  of  contact  is  measured  by  half  the 
intercepted  arc. 

Let  the  tangent  CAB  and  the  chord  A  D  meet  at  the 
point  of  contact  A  ;  then  will  the  angle  B  A  D  be 
measured  by  half  the  intercepted  arc  A  D. 

For  draw  the  diameter  A  E  F. 

Because  A  B  is  a  tangent,  and  A  E  a  radius  at  the 
point  of  contact,  the  angle  B  A  F  is  a  right  angle, 
and  is  measured  by  the  semicircle  A  D  F. 

Because  the  angle  F  A  D  is  at  the  circumference,  it 
is  measured  by  half  the  arc  D  F* 


THEOREMS   ILLUSTRATED. 


153 


Then  the  difference  between  the  angles  B  A  F  and 
D  A  F,  or  B  A  D,  must  be  measured  by  half  the 
difference  of  the  arcs  A  D  F  and  D  F,  or  A  D  ; 

That  is,  the  angle  B  A  D  is  measured  by  half  the 
arc  A  D. 


PROPOSITION    XXIII.     THEOREM. 

DEMONSTRATION. 

We  wish  to  prove  that 

A  tangent  and  cJiord  parallel  to  it  intercept  equal  arcs 
of  the  circumference. 

Let  A  B  be  tangent  to  the  circumference  at  the 
point  D,  and  let  c  F  be  a  chord  parallel  to  the 
tangent ;  then  will  the  intercepted  arcs  c  D  and 
D  F  be  equal. 


154  FIRST  LESSONS  IN  GEOMETKY. 

For  from  the  point  of  contact  D;  .draw  the  straight 

line  D  c. 
Because  the   tangent  and  chord  are  parallel,  the 

interior   alternate  angles  ADC  and  D  c  F  are 

equal. 
But  the  angle  ADC,  being  formed  by  the  tangent 

D  A  and  the  chord  D  c,  is  measured  by  half  the 

intercepted  arc  D  c ; 
And  the  angle  D  c  F,  being  at  the  circumference,  is 

measured  by  half  the  arc  on  which  it  stands,  D  F  : 
Then,  because  the  angles  are  equal,  the  half  arcs 

which  measure  them  are  equal,  and    the    arcs 

themselves  are  equal. 


THEOREMS  ILLUSTRATED.  155 

C          A 


PROPOSITION   XXIV.     THEOREM. 

DEMONSTRATION. 
We  wish  to  prove  that 

The  angle  formed  by  the  intersection  of  two  chords  in 
a  circle  is  measured  by  half  the  sum  of  the  inter- 
cepted arcs. 

Let  the  chords  A  B  and  c  D  intersect  each  other  in 
the  point  E  ;  then  will  the  angle  B  E  D  or  A  E  c 
be  measured  by  half  the  sum  of  the  arcs  A  c,  B  D. 

For  from  the  point  c  draw  c  F  parallel  to  A  B. 

Because  the  chords  A  B  and  c  F  are  parallel,  the 
arcs  A  c,  B  P,  are  equal. 

Add  each  of  these  equals  to  B  D,  and  we  have  B  D 
plus  A  c  equal  to  B  D  plus  B  F;  that  is,  the  sum 
of  the  arcs  B  D,  A  c,  is  equal  to  the  arc  F  D. 

Because  the  chords  A  B,  c  F,  are  parallel,  the  oppo- 
site exterior  and  interior  angles  D  E  B,  D  c  F,  are 
equal. 


156 


FIRST  LESSONS  IN  GEOMETRY. 


But  D  c  F  is  an  angle  at  the  circumference,  and  is 
therefore  measured  by  half  the  arc  F  D. 

Then  the  equal  angle  DEB  must  be  measured  by 
half  of  the  arc  F  D,  or  its  equal  B  D,  plus  A  c. 


PROPOSITION    XXV.     THEOREM. 

DEMONSTRATION. 
We  wish  to  prove  that 

The  angle  formed  by  two  secants  meeting  without  a 
circle  is  measured  by  half  the  difference  of  the  in- 
tercepted arcs. 

Let  the  secants  A  B,  A  c,  intersect  the  circumference 
in  the  points  D  and  E  ;  then  will  the  angle  BAG 


THEOREMS  ILLUSTRATED.  157 

be  measured  by  half  the  difference  between  the 

arcs  B  c  and  D  E. 
For  from  the  point  D  draw  the  chord  D  F  parallel 

to  E  c. 
Because  A  c  and  D  F  are   parallel,  the   opposite 

exterior  and  interior  angles  B  D  F  and  BAG  are 

equal. 
Because  the  chords  D  F,  E  c,  are  parallel,  the  arcs 

D  E  and  F  c  are  equaf. 
If  from  the  arc  B  c  we  take  the  arc  D  E,  or  its  equal 

F  c,  we  shall  have  left  the  arc  B  F  ; 
But  the  angle  B  D  F,  being  at  the  circumference,  is 

measured  by  half  the  arc  B  F  : 
Then  the  equal  of  B  D  F,  or  B  A  c,  must  be  meas- 
ured by  half  the  arc  B  F,  or  half  the  difference 

between  the  intercepted  arcs  B  c  and  D  E. 


APPENDIX. 


NOTE  A.  —  To  those  teachers  who  think  that  the  line  should  be 
derived  from  a  surface,  and  the  surface  from  a  solid,  the  author  would 
say,  that,  according  to  his  experience,  children  apprehend  the  ideas 
conveyed  by  the  terms  line  and  surface  as  readily  as  they  do  any 
ideas  whatever ;  and  that,  therefore,  there  seems  to  be  no  necessity 
for  extraordinary  care  in  this  case  to  avoid  giving  wrong  impressions.- 

Still,  if  it  be  considered  desirable  in  this  manner  to  derive  lines  and 
surfaces,  it  will  be  apparent  that  all  that  can  be  done  in  the  matter  is 
to  give  such  instruction  only  by  way  of  a  preliminary  lesson. 

NOTE  B.  —  Crooked  and  curved  lines  are  here  treated  of  before 
straight  lines,  because  the  first  two  are  defined  by  means  of  an  affirm- 
ative property,  —  they  do  change .  direction  ;  while  the  last  is  defined 
by  means  of  the  absence  of  one,  —  they  do  not  change  direction.  It  is 
easier  for  a  child  to  comprehend  what  is  than  what  is  not. 

NOTE  C.  —  If  the  pupils  are  old  enough,  they  may  be  shown  that 
vertical  lines  cannot  be  parallel,  but  only  seem  so  on  account  of  their 
shortness  and  nearness  to  each  other. 

NOTE  D.  —  This  definition  may  be  considered  objectionable  because 
rhomboid  means  like  a  rhomb.  That  the  more  general  figure,  the 
rhomboid,  has  been  named  from  the  more  restricted  one,  the  rhomb,  is 
unfortunate,  because  it  interferes  with  the  symmetry  of  the  nomencla- 
ture. The  rhomb  possesses  all  the  properties  of  the  rhomboid,  and 
should,  therefore,  when  these  are  considered,  be  called  by  the  same 
name ;  its  additional  property  entitles  it  to  a  name  which  should 
comprehend  the  other  names.  If  the  rectangle  had  been  called  a 
squaroid,  the  difficulty  would  have  been  repeated. 

NOTE  E.  —  If  teachers  consider  it  desirable,  they  may  require  the 
class  to  prove,  by  way  of  corollary,  such  propositions  as  assert  the 
parallelism  of  the  lines  when  the  interior  alternate  angles  are  equal, 
when  the  opposite  exterior  and  interior  angles  areecgalj^i^hi  con" 
tinuation  of  what  has  already  been  done. 


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LD  21-95w-7,'37 
vECOEDS. 

BRADBURY'S  SCHOOL  MUSIC  BOOKS,  &c. 

05208 


SPENCERIAN  SYSTEM  OF  BUSINESS  WRITING. 

THE  NEW  STANDARD  EDITION  OF  THE 

SPENCERIAN  COPY-BOOKS 

REVISED,  IMPROVED,  AND  NEWLY  ENLARGED. 

IN  FOUR  DISTINCT  SERIES. 

COMMON  SCHOOL  SERIES.    Nos.  1,  2,  3,  4,  and  5. 
BUSINESS  SERIES.    Nos.  6  and  7. 

LADIE&  SERIES.     Nos.  8  and  9. 

EXERCISE  SERIES.     Nos.  10,  -11,  and  12. 

The  particular  points  of  excellence  claimed  are 
I.     SIMPLICITY.  2.     PRACTICABILITY.  8.     BEAUTY. 

•$SHIP. 
B  DRAWING 


U.C.  BERKELEY  LIBRARIES 


1ST  SYSTEM 

ctical, 

HEL  PENS. 

not  found  in 
v  VAN 


VCTURED. 

UNIVERSITY  OF  CALIFORNIA  LIBRARY 

-  - ~.»-»».«..u«Nirii«w»«P. -purchasing  a?iy 

"Speaeerian  "  Tens  which  have  not  our  initials,  "I.,  P.,  B.  &  Co.," 
or  "Ivison,  Phinney  &  Co.,"  on  each  Pen. 

These  PENS  will  be  sent  by  Mail  to  any  address  in  the  United  States,  postage 
paid,  on  receipt  of  price  annexed. 


No.  1 per  gross  $1  50 

No.  2 

No.  3 

No.  4 

No.  5 

No.  6 

No.  T 


1  50 
1  50 
1  50 
1  50 

1  50 

2  00 


No.    8 per  gross  $1  50 


No.  9. 
No.  10.. 
No.  11  . 
No.  1-3.  . 
No.  13.. 
No.  14.. 


1  50 

2  00 
2  00 
2  75 

1  50 

2  00 


Sample  gross,  4  kinds  assorted,  excepting  No.  12,  $2  00. 


SAMPLE  CARDS  containing  all  the  Fourteen  Numbers,  PRICE  TWENTY- FIVE  CENTS. 
A  LIDERAL  DISCOUNT  TO  THE  TRADE. 

N.  B. — One  of  the  strongest  proofs  of  the  great  popularity  of  these  Pens,  and  an 
undeniable  confession  to  their  superiority,  is  that  no  less  than  ten  firms  have  man- 
ufactured, or  caused  to  be  made,  Pens  similar  in  style,  under  different  names,  for  which 
they  claim  the  same  qualities  and  favor  as  the  still  unapproached  SPENCERIAN. 

^jr"  Teachers  and  Superintendents  are  invited  to  send  for  our  Catalogue  or  Circulars. 
ADDRESS  THE  PUBLISHERS, 

IVISON,  PHINNEY,  BLAKEMAN  &  CO. 

P.  0.,  Box  1478,  New  York. 


